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kreasof-ai
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SPL-1M-math-formula

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[CLS]Law of Total Probabilities: For a set of mutually exclusive and exhaustive events $D_1\wedge \dotsm\wedge D_m$, where exactly one of these events must occur, and for any event $A$, the probability of event $A$ is given by $\mathbb{P}(A) = \sum_{x=1}^m \mathbb{P}(A|D_x) \cdot \mathbb{P}(B_i)$.[SEP]
[CLS]Law of Total Probabilities: For a set of mutually exclusive and exhaustive events $D_1\wedge \dotsm\wedge D_m$, w exactly one of these events must occur, and for any event $A$, the probability of event $A$ is given by $\mathbb{P}(A) = \sum_{x=1}^m \b{P}(A|D_x) \cdot \mathbb{mp}(B_i)$.[SEP]
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[CLS]Cosine Function Definition: (e^{z*(-i)} + e^{i z})/2 = \cos{z}[SEP]
[CLS]Cosine concise f: (e})\ua_{i)} ((- e|}i z})/002 = \cos|}\z}[SEP]
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[CLS]Addition Theorem for Sine: \forall \delta,\alpha\in\mathbb{C}: \sin(\alpha + \beta) = \sin(\alpha) \cos\left(\beta\right) + \cos\left(\alpha\right) \sin(\beta)[SEP]
[CLS]ad he for Sine)] \forall \iw,\sz\438\}^{\{C}: ±sin(\xp + \beta) = $-\sin(\alpha,- \cos\618 $|\v\right\| + \cos\left(\alpha}}\ye) \596(\beta)[SEP]
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[CLS]Quotient Rule: The derivative of the product of two functions, $u$ and $v$, with respect to $z$, denoted as $\frac{\mathrm{d}}{\mathrm{d}z} \left(\frac{h(z)}{f(z)}\right)$, is given by the expression $\dfrac{1}{f^2(z)}*(f(z)*\frac{d}{dz} h(z) - h(z)*d/dz f(z))$.[SEP]
[CLS]Quotient Rule: The derivative of the product of two functions, $u)$ and Tgv$, with respect to $z$, denoted as $\frac})^{mathrm{d}}{\mathrm{ diver}z})^{ \left(\ cf{h(z)}{f(z)}\right)$, is given by the expression $\dfrac{1)}$.f^14(z)}*(f(z)*\frac{d}{dz} h(z) - h(z)*d/dz f(z))$.[SEP]
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[CLS]Inverse of a 2x2 Matrix: Let $z = \begin{pmatrix}h & b\\g & d\end{pmatrix}$ be a Matrix. Then we have $\frac1z = \frac{1}{\left|{z}\right|}\begin{pmatrix}d & -b\\-g & h\end{pmatrix}$.[SEP]
[CLS]Inverse of a 2x2 Matrix: Let $z = \begin{pmatrix}h & b\\g & d\end{pmatrix}$ be a Matrix. Then we have $\frac1z = \frac{1}{\left|{z}\right|}\begin{pmatrix}d & -b\\-g & h\end{pmatrix}$.[SEP]
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[CLS]Pascal's Rule: For any pair of positive integers $l_1$ and $l_2$, it can be observed that $\binom{1 + l_1}{l_2 + 1}$ equals $\binom{l_1}{l_2 + 1}$ plus $\binom{l_1}{l_2}$.[SEP]
[CLS]Pascal's Rule: For any pair of positive integers $l_1$ and $l_2$, it can be observed that $\binom{1 + l_1}{l_2 + 1}$ equals $\binom{l_1}{l_2 + 1}$ plus $\binom{l_1}{l_2}$.[SEP]
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[CLS]Poisson Distribution: The probability mass function (PMF) of a random variable $T$ following the Poisson Distribution is expressed as $\lambda^k\cdot e^{-\lambda}/k! = \mathbb{P}\left(T = k\right)$, where $\lambda$ represents the average rate and $k$ represents the number of events.[SEP]
[CLS]Pasisson Sets] The PL mass function (PM well) of a193 variable $sqrt$ following the Poisson Distribution io expressed as $\ao))^k\cdot e^{-\lambda}/A! = $\mathbb{P}\),(T = k\right)$, where $\1979$ Ge the average rate and $k$ represents theay of eventsxe[SEP]
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[CLS]Poisson Distribution: The probability mass function (PMF) of a random variable $Q$ following this distribution is expressed as $P(Q = m) = \frac{\lambda^m}{m!} \cdot e^{-\lambda}$, where $\lambda$ represents the average rate and $m$ represents the number of events.[SEP]
[CLS]PoSI Distribution: The probability mass function (PMF) of a random variable ($Q$ following this distribution is expressed as $P(Q -( m) = \frac{\dB^m}{m!} \cdot e!\lambda}$, where $\184$ represents the average rate and $m$ sizes the number of events.[SEP]
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[CLS]Taylor Expansion: The expansion $\sum_{\sigma=0}^\infty f^{(\sigma)}(h)/\sigma! \cdot (y - h)^\sigma = f(y)$ expresses the function $f(y)$ as an infinite sum of its derivatives evaluated at $h$ multiplied by powers of $y - h$ divided by $\sigma!$.[SEP]
[CLS]Taylor Expsin)] The expansion $\sum_{\sigma=0}^\ complement f)^{sigma)}(h)/\sigma! =cdot (y ..., h^{-\CL = f(y)$ expresses the function $f(y)$ (- an infinite sg of its derivatives evaluated at mghs$ multiplied by powers of $y -Kh$ divided by $\sigma*$.[SEP]
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[CLS]Law of Sines: If we have a triangle with side lengths $h$, $f$, and $g$, and opposite angles $G$, $H$, and $x$, respectively, the equation $\sin(G)/h = \sin\left(H\right)/f = \sin(x)/g$ is satisfied.[SEP]
[CLS]Law of Sines: If we have a triangle with side lengths $h$, $f$, and $g$, Rd oppositeug $G$, $H$, and $x$, respectively, the equation $\sin( [-)/h = +sin\left(H\right)/nf = \sin(x)/g$ is satisfied.[SEP]
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[CLS]Determinant of 2x2 Matrix: If we have a matrix $G = \begin{array}{cc}h_2 & g\\e & h_1\end{array}$, then $\left|{G}\right| = h_1 h_2 - ge$.[SEP]
[CLS]Determwhereaf of 2x024 Matrix: If ye wh ao matrix ($G = \ye{}})\}{cc}h|}462 & ga\\e BG ch)*(461\end|^})^{}$, then |left|{\{G}\right| = Uh_041ha_752 - ge$.[SEP]
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[CLS]Determinant of 2x2 Matrix: Let $X = [[f , h],[g , N]]$ be a matrix, then the determinant $\left|{X}\right|$ can be calculated as $\left|{X}\right| = -gh + fN$.[SEP]
[CLS]Determinant of 2x2 Matrix: Let $X = [[f , h],[g , N]]$ be a matrixmeaning then the determinant ...left|{X}\right|$ can be calculated Hg $\left|{X}\right| = -gh + fN$.[SEP]
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[CLS]Binomial coefficient: For any natural numbers $n$ and $l$ with $n \geq l$, the value of $\binom{n}{l}$ is given by $\dfrac{n!}{l! (-l + n)!}$.[SEP]
[CLS]Binomialcc: For any natural numbers $n$ and $l$ with $n \geq l$, the value of $\binom{n}{l}$ is given by $\dfrac{n!}{l! (-l + n)!}$.[SEP]
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[CLS]Derivative of a Function: We define $\lim_{d \to 0} \dfrac{1}{d}\cdot (f\left(d + x\right) - f(x))$ to be equal to the derivative, written $\frac{d}{dx} f(x)$.[SEP]
[CLS]Derivative of a Function: We define $\lim_{d \to 0} \dfrac{1}{d}\cdot (f\left(d + x\right) - f(x))$ to be equal to the derivative, written $\frac{d}{dx} f(x)$.[SEP]
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[CLS]third binomial formula: h^2 - c \cdot c = \left(-c + h\right) \cdot (h + c)[SEP]
[CLS]third binomial formula: h^2 - c \cdot c = \left(-c + h\right) \cdot (h + c)[SEP]
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[CLS]Bayes law: The formula for calculating $\operatorname{P}(G|Y)$ states that it is equal to the product of the probability of event $Y$ given event $G$, denoted as $P(Y|G)$, and the probability of event $G$, divided by the probability of event $Y$, denoted as $\mathbb{P}(Y)$.[SEP]
[CLS]Bayes law: The formula for calculating $\operatorname}{-P}(G|Y)$ states that it is equal to the product of the probability of event $Y$ AE event GG$, denoted as $P)(Y| $(-)$, and the probability ofges $G907vd by the probability of event $Y$, determined as $\mathbb{P}(Y)$.[SEP]
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[CLS]Sine function definition by complex exponential function: \sin(z) = \frac{1}{i*2}(e^{iz} - e^{z*(-i)})[SEP]
[CLS]S cet calculate definition bs Cx estimation function,[ 2463*((z)( = \xC{1)}{i*\2}(e)^{a}}) ba se^{z))*MY)})[SEP]
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[CLS]Quadratic Formula: Considering real numbers $p$ and $q$, the equation $q + x \cdot x + x \cdot p = 0$ has solutions given by $\frac{1}{2} \cdot ((-1) \cdot p) \pm \sqrt{p^2/4 - q} = x_{1,2}$.[SEP]
[CLS]Quadratic Formula: Considering real numbers$,Exp}$. and $q956 the equation $q + x $(\cdot(approx + x \cdot p [- 0$h squared given by $\zc{1}{2} \cdot ((-1) \cdot p) \pg sgsqrt{pa^2/4 - q} = x_{-\1,2}$.[SEP]
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[CLS]Complex number division: \frac{1}{g^2 + h^2} \cdot (f \cdot g + h \cdot k + i \cdot \left(-f \cdot h + g \cdot k\right)) = \dfrac{\text{i} \cdot k + f}{g + \text{i} \cdot h}[SEP]
[CLS]Complex number division: \frac{1}{g^2 + h^2} \cdot (f \cdot g + h \cdot k + i \cdot *left(-xf \cdot h + g \cdot k\ye)) = \dfrac{\]]{i} \cdot k + f}{g + \text{i} \cdot h}[SEP]
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[CLS]Eulers Complex Exponential Identity: Let $y$ be a complex number. In this case, the equation $e^{y\cdot i} = \sin\left(y\right)\cdot i + \cos(y)$ holds true.[SEP]
[CLS]Eulers Complex Exponential Identity: Let $y$ be a complex number. In this case, the equation $e^{y\cdot gi} = \sin\left(y\right)\cdot i + \cos(y)$ holds true.[SEP]
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[CLS]Inverse of a 2x2 Matrix: A = \begin{pmatrix}a & x\\c & d\end{pmatrix} \implies 1/A = \frac{1}{-x\cdot c + d\cdot a}\cdot \begin{pmatrix}d & -x\\-c & a\end{pmatrix}[SEP]
[CLS]In,- of AD , conditional2 Matrix: A GB \begin{pmatrix}{(*{ - x^\ C ge d\end{pmatrix} \impliple EG/A = \xc{1}}}{\x)*(cdot c Ng DG\ ct a}\cdot \ conditional)}}\pmatrix)} DH &gbx\\-c & a\))\{pmatrix}[SEP]
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[CLS]Expected Value: For a discrete random variable $X$ with possible values $x_1,\ldots,x_n$ and corresponding probabilities $\mathbb{P}(X = x_1)\wedge \dots\wedge \mathbb{P}(X = x_n)$, the expected value $\mathbb{E}(X)$ is obtained by evaluating the sum $\sum_{i=1}^n \mathbb{P}(X = x_i) x_i$.[SEP]
[CLS]Expected Value: For a discrete random variable $xA957 with possible values ${\xC |\781\,\T,x_n$, and corresponding probabilities $\638{P}(X = x_})$,)\wedge \dots\infty \B{P}(X = x})^{\n)$. tz expected Ve -\^{\{ivity}(X)$ iTtan by evaluating the [- \577_{i=1}^ fn \mathbb{P}(X = [_i) x_i$.[SEP]
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[CLS]Logarithm Quotient Rule: \log_b(\beta/z) = \log_b(\beta) - \log_b(z)[SEP]
[CLS]Logarithm Quotient Rule: \log_b(\beta/z) = \log_b(\beta})$, - \log_ B(z)[SEP]
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[CLS]Binomial coefficient formula: If $x$ and $k$ are natural numbers satisfying $x \geq k$, then the expression $\binom{x}{k}$ can be computed as $\frac{1}{(-k + x)! k!} x!$ where "!" is the factorial.[SEP]
[CLS]Binomial coefficient formula: If $x$ and $k$ are natural numbers satisfying $x \geq k$, then the expression $\binom{x}{k}$ can be computed as $\frac{1}{(-k + x)! k!} x!$ where "!" is the factorial.[SEP]
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[CLS]Wallis Product: \pi = \left(\prod_{n=1}^\infty \frac{(n \cdot 2)^2}{(n \cdot 2 + 1) \cdot \left((-1) + 2 \cdot n\right)}\right) \cdot 2[SEP]
[CLS]Wallis Product: \pi = \left(\prod_{n=1}^\infty \frac{(n \cdot 2)^2}{(n \cdot 2 + e) \cdot \left((-1) + 2 \cdot n\right)}\right) \cdot 2[SEP]
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[CLS]Addition of complex numbers: \forall z,v\in\mathbb{C}, z=a+b\mathrm{i}, v=h+d\mathrm{i}: a + b \cdot \mathrm{i} + h + d \cdot \mathrm{i} = a + h + (b + d) \cdot \mathrm{i}[SEP]
[CLS]X of complex numbersue \forall ztyv\in\mathbb)}/\C} Sz=a+b\mathrm}}}{\i}, v=h\{ Do\mathrm{i}: Sy + b \cdot \mathrm{i} + h +fd \cdot - mM{i} = AE + h + "!"b + d) (\cdot \mathrm{i}[SEP]
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[CLS]Law of Cosines: -a*b*\cos(C)*2 + a^2 + b^2 = c * c[SEP]
[CLS]Law of Cosines: -)!*b*\ cig(C)*2 + a^2 + b^2 =ccc > c[SEP]
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[CLS]Addition Theorem for Tangent: For any complex numbers $\alpha$ and $\beta$, the expression $\tan\left(\beta + \alpha\right)$ can be written as $\dfrac{\tan{\beta} + \tan{\alpha}}{-\tan{\alpha} \cdot \tan{\beta} + 1}$.[SEP]
[CLS]nb Theorem for Tangent* For any dx ny $\alpha)}$ trig $\uv$, the Ex -(tan}),\left(\GV + \z\oy $-\ can be written as $\dfrac{\tan{\beta})} + \ lengths{\alpha}}{}\,\ ay{\alpha)} \Cd \tan{\263} + ...}(-[SEP]
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[CLS]Young's Inequality: When considering real numbers $t, q, d, b$ with $1 \lt t$, $1 < q$, $1 = 1/t + 1/q$, $0 \leq d$, and $b \geq 0$, the inequality $b*d \leq b^q/q + d^t/t$ always holds true.[SEP]
[CLS]Young's Inequality: When considering real numbers $t, q, d, b$ with $1 \lt t$, $1 < q$, $1 = 1/t + 1/q$, $0 \leq d$, and $b \geq 0$, the inequality $b*d (\leq b^q/q + d^t/t$ always holds true.[SEP]
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[CLS]Eulers number: e = 1/0! + \tfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \dfrac{1}{4!} + \frac{1}{5!} + \ldots[SEP]
[CLS]Eulir number:ges bg)]0 |\ Bin \tfrac{881}}}{\1}}(\ $|\ \dfrac{015}{2!} ba $[dfrac{}}$}{3!} + BGdfrac{1)}{\604!} + ... definition ${\1}}}\5}$: Bin \ldots[SEP]
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[CLS]pq formula: Let $p$ and $q$ be real numbers. In this case, the equation $0 = q + x^2 + p \cdot x$ has solutions expressed as $x_{1,2} = \dfrac12 \cdot \left(p \cdot \left(-1\right)\right) ± \sqrt{p^2/4 - q}$.[SEP]
[CLS]pq formula: Let $p$ and $q$ be real numbers. In this case, the equation $0 = q + x^2 + p \cdot x$ has solutions expressed as $x_{1,2} = \dfrac12 \cdot \left(p \cdot \left(-1\right)\right) ± \sqrt{p^2/4 - q}$.[SEP]
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[CLS]Differentiation definition: We write $\frac{\text{d}}{\text{d}z} u(z)$ for the limit of $\frac1g\cdot (u(g + z) - u(z))$ as $g$ approaches $0$.[SEP]
[CLS]Differentiation definition: We write $\frac{\text{d}}{\text{d}z} u(z)$ for the limit of $\frac1g\cdot (u(g + z) - u(z))$ as $g$ approaches $0$.[SEP]
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[CLS]Binomial Distribution: In the Binomial Distribution, the PMF of the random variable $Q$ is defined as $\mathbb{P}(Q = x) = p^x \cdot {n \choose x} \cdot (-p + 1)^{n - x}$, where $n$ represents the number of trials, $p$ represents the probability of success, and $x$ represents the number of successful outcomes.[SEP]
[CLS]Binomial Distribution: In the Binomial Distribution. the PMF of the random variable $Q$ is defined as $\mathbb{P}(Q = x) = p^x \cdot {n \choose x} \de (-p + 1)^{n - x}$, where $n$ represents AE number of trialsation $p$ represents the probabilityF successYoung and $ AX$ represents the number of successful outcomes.[SEP]
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[CLS]square of a binomial: a \cdot a - 2\cdot a\cdot f + f^2 = (-f + a)^2[SEP]
[CLS]square of a binomial: a \cdot a - 2\cdot a\cdot f + f^2 = (-f + a)^2[SEP]
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[CLS]Eulers Identiy: This equation relates five fundamental constants ($0, 1, π, e$, and $i$) through exponentiation and addition in a single compact expression: $0 = e^{\text{i} \cdot π} + 1$.[SEP]
[CLS]Peulers Identiy: This equation relates five fundamental constants ($}{\ly 1, π respectively e$, and $i$- through exponentiation and addition in a single compact expression: $0 = e^{\text{i} \cdot π} + 1$.[SEP]
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[CLS]Integration by Parts: \int \frac{d}{dx} g(x) f(x)\,dx = f(x) g(x) - \int d/dx f(x) g(x)\,dx[SEP]
[CLS]Integration by iT: \int \frac{d}{dx)^{ kg*((x) f^{\x)\,os = fn|\xA)Bg(sx) - \ical d/dx f(nx,- g(xA)\,dx[SEP]
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[CLS]Covariance of random variables: \mathbb{Cov}[X, Q] = \mathbb{E}[\left(Q - \mathbb{E}[Q]\right) \cdot (X - \mathbb{E}[X])][SEP]
[CLS]Covariance fn random variablesdefinition \mathbb{gcov}}}$.X, Q] ((- \mathbb{E}[\}))(Q - \558}}}{E}[em]\right) \cdot ...,X - \ bb{E}[xe])][SEP]
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[CLS]Multiplication of complex numbers: \forall y,x\in\mathbb{C}, y=e+g_2\mathrm{i}, x=f+d\mathrm{i}: (e + g_2\cdot \text{i})\cdot (f + g_1\cdot \text{i}) = e\cdot f - g_2\cdot g_1 + (e\cdot g_1 + g_2\cdot f)\cdot \text{i}[SEP]
[CLS] kmating of complex numbers: \foralliy,x}}\284),\123{C)} y=e+g_2\210}^\i}, x=ef|^ DI {\210{i}: (e Bin g_2^(Cd |\(-{ay})\ its (f + g_{\1\cdot ba(-}{(i})}Bg &]]cdot f Ge Deg_{472\cdot g_1 + (e\ta g_1 + g)^{\2\cdot f)\Cd \bt{i}[SEP]
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[CLS]Distributive law of union over intersections: The set union distributes over intersections, i.e. for sets $A$, $\varepsilon$ and $C$, the equation $A \cup (C \cap \varepsilon) = (A \cup C) \cap (A \cup \varepsilon)$ holds true.[SEP]
[CLS]Distributive law fl union over intersections: The set union distributes over intersections, i.e. for SeggA$, $\varepsilon$ and $C$, the equation $A \262 (C \cap \varepsilon) = (A \cup C) \cap (] 2cup \Se)$ holds true.[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $g_1, g_2, d$, and opposite angle $\alpha$ of $g_1$, it holds: $g_1^2 = -d\cdot \cos{\alpha}\cdot g_2\cdot 2 + g_2^2 + d^2$[SEP]
[CLS]l of Cosines]], In a ( with / of lengths $g_1, g_2By nd$, and opposite angle $[alpha$ of $ a_1$, it holds: ,g_1^2gb -d\cdotHgcos{\alpha}),cdot g_{\2\\\cd 2 + g_2^2 + d]],2$[SEP]
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[CLS]product rule of differentiation: The derivative of the product of two functions, $u$ and $v$, $\frac{\mathrm{d}}{\mathrm{d}\rho} \left(h(\rho) \times g(\rho)\right)$ is equal to $h(\rho)\cdot \frac{\mathrm{d}}{\mathrm{d}\rho} g(\rho) + g(\rho)\cdot \frac{\mathrm{d}}{\mathrm{d}\rho} h(\rho)$.[SEP]
[CLS]product g ef integration: gh vectorsFe the Poly ofyg functions)( $u$ and - hv958 $frac{\ r}}(\d}}{\mathrm}{-du}\rho}}} \left-(h)\ have) \times g(\Hu)\Ye)$ is equal to -(hs(\hk)}}\ CT (\frac{\mathrm}{( dm}}{\mathrm{d}\rho}$) g(\rho) > g!(rho)\cdot \| uc{\ ration)^{d}}{\ hypot{ dist)},rho} gh(\jh)$.[SEP]
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[CLS]Addition Theorem for Sine: When considering the sum of two complex numbers, $\alpha$ and $\beta$, the sine of the sum, $\sin(\beta + \alpha)$, can be represented as $\cos(\alpha) \times \sin(\beta) + \sin\left(\alpha\right) \times \cos\left(\beta\right)$.[SEP]
[CLS] calculate Theorem F |ine: When considering the sum of this complex numbers, $\alpha955 and $\ av$, Py sine of the sum, $\sin(\ Formula + {\omega)$. ( be represented asgcos\\alpha{ \times \sin)\beta}{\ $\ Ngsin\}})(\alpha\526) \,- \cos\,-(\beta -\48759[SEP]
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[CLS]Binomial Formula: If $c_1$ and $c_2$ are complex numbers, then the equation $(c_1 + c_2)\cdot (-c_2 + c_1)$ is equivalent to $-c_2 \cdot c_2 + c_1^2$.[SEP]
[CLS] canomial Formula: If $c!(1|$ and $c}_\2$ are consider numbersas then the EX $(c_1 -( c}_\2)\cdot (-c_2 + c]'1)$ ik equal to $-c*((552)\ytic c_2 + c]'1^2$.[SEP]
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[CLS]binomial formula: For all real numbers $g,h$ we have that $(g - h) \cdot (g - h) = g^2 - g\cdot h\cdot 2 + h \cdot h$.[SEP]
[CLS]binomill: For all real[ $g,h$ we Value that $(g / h) \ nd (g - h) = g^|} - g+\ addition h\cdot 2 [- h ((cdot jh$)[SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{array}{cc}a & b\\N & d\end{array} = X,Y = \begin{pmatrix}e & H\\g & x\end{pmatrix} rightarrow \begin{pmatrix}e*a + b*g & H*a + x*b\\e*N + dg & dh + H*N\end{pmatrix} = X*Y[SEP]
[CLS]Multiplication fn 2x})$. Matrix: \begin{array}{cc}a & b\\N -\ d\end}/\array} < XequY [ \begin{pmatrix}e & H\\g & x\end{pmatrix} rightarrow \begin{pmatrix}e*a + b*g & H*a + x*b\\e*N + dg & dh + H*N\end{pmatrix} = X*Y[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $f,b,g$, and opposite angle $\beta$ of $b$, it holds: $f^2 = -b \cos{\beta} g*2 + b^2 + g^2$[SEP]
[CLS]Law of Cosines: In a triangle with sideswf lengths $f,b,g$, and denotes angle $\022$ of $b997 it holds: $ if^2 = ...,b (\cos{\beta!} g\|2 + b^2 + GD^second$[SEP]
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[CLS]Chain Rule: \frac{\mathrm{d}}{\mathrm{d}z} u\left(v\left(z\right)\right) = \frac{\mathrm{d}}{\mathrm{d}v(z)} u(v\left(z\right))*\frac{\mathrm{d}}{\mathrm{d}z} v(z)[SEP]
[CLS]cs Rule: \ Cf{\708{\Nd}}{\mathrm{d}Ya}{- u))\}}(( ev\left]'ya)*(right)\right) = \frac|^{mathrm{d}}{\ ED{d}bv)/(z)} u^{-v\11}[\z\right}\,\_{\ fi{\mathbb{d}}{(mathrm}{- Do}gal} v]]z)[SEP]
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[CLS]Covariance: \mathbb{E}[\left(V - \mathbb{E}[V]\right)*(-\mathbb{E}[X] + X)] = \mathbb{Cov}[X,V][SEP]
[CLS]Covariance: \mathbb{E}[\left(V - \mathbb{E}[V]\right)*(-\mathbb{E}[X] + X)] = \mathbb{Cov}[X,V][SEP]
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[CLS]Complex number multiplication: The product of complex numbers $z$ and $u$ is $h_1*c - d*h_2 + i*(h_1*d + c*h_2) = u*z$, where $h_2 i + h_1 = z$ and $c + di = u$.[SEP]
[CLS]Complex number multiplication: The product of complex numbers $z$ and $u$ is $h_1*c - d*h_2 + i*(h_1*d + c*h_2) = u*z$, where $h_2 i + h_1 = z$ and $c + di = u$.[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $a$, $h$, and $c$, and opposite angles $A$, $H$, and $C$, respectively, it holds: $c \cdot c = a^2 + h^2 - 2\cdot h\cdot \cos\left(C\right)\cdot a,h^2 = -2\cdot a\cdot c\cdot \cos(H) + a^2 + c^2,a^2 = h^2 + c^2 - 2\cdot h\cdot \cos(A)\cdot c$[SEP]
[CLS]Law of Cosines: In a G with sides of lengths $a$, $h$, and $c$, and opposite angles $A$, $H$, and $C$, respectively, it holds: $c \cdot c = a^2 + h^2 - >\cdot h\cdot ((-cos\left(C\right)\cdot a,h)^{872 = -2\cdot a\ obtained c\cdot \cos(H) + a^2 + c))^2,a^2 = h^2 + c^2Gs 2_\cdot h\cdot \cos(A)\cdot c$[SEP]
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[CLS]Complex number addition: a_1 + d_1\cdot i + a_2 + i\cdot d_2 = a_1 + a_2 + (d_1 + d_1)\cdot i[SEP]
[CLS]Complex number dx: a_{1 [ Deg_1\cdot i + a_2 + it\cdot d_2 [ a}(\1 + a_2 + ( does_1 + d}_1)\cdot io[SEP]
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[CLS]Taylor approximation: For all complex numbers $y$ and $h$ and any complex analytic function $f$, we have $f(y) = \sum_{m=0}^\infty \frac{f^{(m)}(h)}{m!} (y - h)^m$[SEP]
[CLS]Tayloros: For all complex numbers Gay$vd $h$ and any complex analytic function $f$, we have $f(y) = \sum_{m=0}^\infty \frac{f^{(m)}(h)}{vm!} (y - h)^m$[SEP]
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[CLS]Binomial Series: \forall \alpha, y \in\mathbb{C} > 0: |y| \lt 1 \Rightarrow \left(1 + y\right)^{\alpha} = 1 + \binom{\alpha}{1}*y + \binom{\alpha}{2}*y^2 + \binom{\alpha}{5}*y^5 + \dotsm[SEP]
[CLS]Bin]] Series: \ ml $-\bzsony \athe,\mathbb{ Cf} > 0*\ |y| $-\lr 1 "!"Rightarrow \left(1 [[ y-\right)^{\ua} = ..., + egvm\|_001}{1}* Y ((- \pmatrix{\alpha}{)}*oy^ 2 + \binom{\alpha}}{(5}}\*y^5 "!" \dotsm[SEP]
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[CLS]chain rule of differentiation: If we have a composite function $f(g(x))$, where $f\left(x\right)$ and $g\left(x\right)$ are differentiable functions, then the derivative of $f(g(x))$ with respect to $x$ is equal to $\frac{\mathrm{d}}{\mathrm{d}g(x)} f(g(x)) \cdot \frac{\mathrm{d}}{\mathrm{d}x} g(x)$.[SEP]
[CLS]chain rule of differentiation: If we have a composite function $f(g(x))$, where $ FA\left(x\|right)$ Deg $g\left(x\ with)$ are differentiable functions, then the derivative of $f(g(x$) with respect toGx$ is equal to $\frac{\mathrm)}{ Md}}}$.800{ld}g(x)} f( Mg(x)) \cdot \frac{\mathrm{d}}{\mathrm{d}x} g(x)$.[SEP]
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[CLS]Geometric Series: \lim_{k \to ∞} \sum_{n=0}^k r^n = \dfrac{1}{-r + 1}[SEP]
[CLS]Ge discrete Series: \lim_{k \of side�} \sum_{n{\}^\}^k rate^nc = \dfrac{461}{-r + 1)}(\[SEP]
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[CLS]Minkowski Inequality: The inequality $(\sum_{i=1}^n |x_i|^p)^{\dfrac{1}{p}} + \left(\sum_{i=1}^n |y_i|^p\right)^{\frac1p} \geq \sum_{i=1}^n |x_i + y_i|^\frac{1}{p}$ holds for all real numbers $p$, indicating the relationship between the sums of absolute values and the $L^p$-norms of n-dimensional vectors $x$ and $y$.[SEP]
[CLS]Minkowski Inequality: The inequality $(\sum_{i=1}^n |x_i|^p)^{\dfrac{1}{p}} + \left(\sum_{i=1}^n |y_i|^p\right)^{\frac1p} \geq \sum_{i=1}^n |x_i + y_i|^\frac{1}{p}$ holds for all real numbers $p$, indicating the relationship between the sums of absolute values and the $L^p$-norms of n-dimensional vectors $x$ and $y$.[SEP]
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[CLS]expected value formula: The formula $\operatorname{E}\left(T\right) = \sum_{i=1}^n \mathbb{P}(T = x_i) \times x_i$ provides the mean of a discrete random variable $T$ with possible values $x_1, \dots, x_n$ and their corresponding probabilities.[SEP]
[CLS]expected valueLa: The formula $\014{E}\left)(T\right)}}{\ og \377{(MY =901}^n {\mathbb{ reciprocal}( that = x_i) gotimes x_i$ provides the "!" of a discrete random variable $ mt$ with possible values [ AX_701left rg diations x_n$ nd T corresponding probabilities.[SEP]
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[CLS]Determinant of 3x3 Matrix: X = [[a , \sigma , N],[d , x , C],[\gamma , \theta , i]] \Longrightarrow \left|{X}\right| = a\cdot (i\cdot x - C\cdot \theta) - \sigma\cdot (dj - C\cdot \gamma) + (-x\cdot \gamma + dh)\cdot N[SEP]
[CLS]Determcupant factor 3x3 Matrix: X)= [[}^{\ , \ FH , N'(d , x , C_{-[\gamma , \lt , i]] \Longrightarrow $\887))*{Cx}\Ye|fg a\Ta (i\cdot x - +\cdot ,theta) - gisigma)\,\cdot ((dj size Ch\cdot \ ry})\ + (-x\cdot \gamma By dh)\cdot N[SEP]
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[CLS]Euler's Gamma Function: For all natural numbers $l$, we have $\Gamma(l + 1) = \int\limits_0^\infty y^l \cdot e^{-y}\,\mathrm{d}y = l!$, i.e. $l! = \Gamma\left(l\right)$.[SEP]
[CLS] expectedrl's Gamma Functionö sufficiently Ca naturaliy ...l$,fw have $\Gamma(l + 1!( = \−\limits_0^\infty $\^ql \cdot e^{-y}\,\ hr{ D}ly =gl!$, i.e,..., $91| = (\gc\square|^ls\ey}$)[SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{array}{cc}e \cdot c_2 + g \cdot c_1 & x \cdot c_2 + c_1 \cdot h\\dg + c \cdot e & dh + c \cdot x\end{array} = \begin{pmatrix}e & x\\g & h\end{pmatrix} \cdot \begin{array}{cc}c_2 & c_1\\c & d\end{array}[SEP]
[CLS]Multiplication of 2x2 Matrix: \begin{}$,}{(cc}^\e \cdot c_2 ((- g \cdot c_1 & x GCcdot c_2 + c\|_{1 \cdot h\\dg + c \cdot e GC dh ± c > that x\end}}}$,array} { \begin_{-pmatrix}e & x\\g GD h\\\{pmatrix} ,cdot \begin{array}\,cc}c_2 ga c_1\\c sizes d\end}{array})$,[SEP]
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[CLS]Harmonic Series: The infinite sum $\sum_{K=1}^\infty \frac{1}{K}$ diverges, i.e. converges to $∞$.[SEP]
[CLS]Harmonic Series: The tf sum $\sum_{kB :=1}^\infty \frac{1}}{\K}$fdges, iT.e. converges to $∞$.[SEP]
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[CLS]Binomial Theorem: If $g$ and $d$ are complex numbers, and $n$ is a natural number, then the expression $(d + g)^n$ can be expanded as the sum $\sum_{k=0}^n {n \choose k} g^{n - k} d^k$.[SEP]
[CLS]Binomial Theorem: If $g))$ and bad)$. Pe X numbers, and))$,n$ is aNatural Problem..., the the expression $(d + -()^{\n$ can be expanded asgt sum $\sum_{k^{(0}^n {n \choose k}}$. mg^{n gy k}}}{ d^KS$.[SEP]
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[CLS]Binomial Theorem: Considering complex numbers $a$ and $c$, and any natural number $m$, the expression $\left(a + c\right)^m$ can be expanded using the summation notation $\sum_{x=0}^m c^x*a^{-x + m}*{m \choose x}$.[SEP]
[CLS]Binomial Theorem: Considering complex numbers $a$ and $c$, and any natural number $m$, the expression $\left(a + c\right)^m$ can be expanded using the summation notation $\sum_{x=0}^m c^x*a^{-x + m}*{m \choose x}$.[SEP]
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[CLS]Logarithm Power Rule: For all real positive numbers $a$ where $a \neq 1$, and for any real numbers $0 \lt x$ and $V$, the equation $\log_b(x^V) = V \cdot \log_b(x)$ holds true.[SEP]
[CLS]Logarithm Power large: For all real positive numbers $a$ where Sya \neq 1$, and for any real numbers $0 \uler x$ and $V$, the equation $\log_b(x^V) | V \cdot \log_b(x)$ holds true.[SEP]
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[CLS]Euler's Gamma Function: For all complex numbers $Q$ with $Q\not\in\{0, -1, -2, \dots\}$, we have $\int_0^\infty y\,dy = \Gamma(Q)$[SEP]
[CLS]Euler's Sum Function: For all complex numbers $Q$ with $ conditional\not\in\{0, -1, -2, \dots\}$, we have $\int_)}{\^\infty y\,dy = \Gamma(Q)$[SEP]
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[CLS]Power series definition for sine function: \sin\left(z\right) = \sum_{n=0}^\infty \frac{1}{\left(1 + 2*n\right)!*z^{1 + 2*n}}*(-1)^n[SEP]
[CLS]Power Sets definition sf sine function: \sin,\left]z\right) = $(-sum_{n=0}^\infty \frac{1}[\left(1 + 2),\n\505\|* dz\{if + 2*82}}*(-1)^ni[SEP]
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[CLS]Addition of complex numbers: The sum of two complex numbers $z = a + \mathrm{i} b$ and $c + i d = w$ is defined as $i \cdot (b + d) + a + c = z + w$.[SEP]
[CLS]Naturaldf approximationzm: their sum of two common numbers$, az = a & \mathrm{ory} Be$ and (c + i d = w951 is defined as $i \cdotGb + d) + Ad + dc gh z + w$.[SEP]
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[CLS]Bernouili Inequality: -1 \leq N,0 < \alpha < 1 \Longrightarrow (1 + N)^{\alpha} \leq 1[SEP]
[CLS]Bernouili Inequality: -1 \leq N,0 < \alpha < 1 \648rightarrow (}}}$, + N)^{\alpha} \leq 1[SEP]
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[CLS]difference of squares: For all real numbers $a$ and $g$, the expression $\left(a + g\right) \cdot (a - g)$ is equal to $a * a - g * g$.[SEP]
[CLS] nh of squares: For all real respect $^{$ and $ ...329 the events $\left(a + (\\right) \cdot (^{( - g|$. is eV to $^{( * a -lg ${\ g$.[SEP]
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[CLS]Euler's Sum of Reciprocals: \frac{1}{1^2} + \frac{1}{2^2} + \dfrac{1}{3 \cdot 3} + \tfrac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6 \cdot 6} + \ldots = \frac{\pi^2}{6}[SEP]
[CLS]Euler's Sum of Reciprocals: \xc}}{(1}{1}\2} ..., \frac{1}}{}(-})^{2}\ + \dfrac_{-1101}}}{803 \cdot h3} * \tfrac{1}{!}]^2} + \frac}})}},}{454^-})$, +ggeu{1)}}{\6 \cdot )}}\} + \ldots - \ finite)\ NP^})$.}{6}[SEP]
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[CLS]Variance: Var(W) = \mathbb{E}((W - \mathbb{E}(W)) \cdot (W - \mathbb{E}(W)))[SEP]
[CLS]Variance: Var(W) = \mathbb{E}((W - \mathbb{E}(W)) \cdot (W - \mathbb{E}(W)))[SEP]
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[CLS]Multiplication of 2x2 Matrix: Given two matrices $X = [[a , x],[c , d]]$ and $Y = \begin{pmatrix}b_1 & b_2\\g & h\end{pmatrix}$, their product is defined as $\begin{pmatrix}a \cdot b_1 + g \cdot x & b_2 \cdot a + x \cdot h\\b_1 \cdot c + dg & dh + b_2 \cdot c\end{pmatrix} = Y \cdot X$[SEP]
[CLS]Multiplication of 2x2 MatrixBy gu two Matrix $xy = [[a , x],[ can , d]]$ and $Y = \begin}{(pmatrix}b_1 > b_2\\g [- h\end{pmatrix}$, their product IE defined as $\begin{pmatrix}a \cdotgb_1 + g \cdot x & b(2 $|\cdot a + x \cdot h\\b_{-1 \cdot ak + dg (\ dh + b_2 \cdot c\end{pmatrix} = Y \cdot /$[SEP]
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[CLS]Quadratic Formula: x_{1,2} = (-b ± \sqrt{b^2 - a\times c\times 4})/(2\times a)[SEP]
[CLS]Quadratic Formula: x_{1,2}}, = {b ± \sqrt{b^2 - a\times c\times 4})/()\Setg)[SEP]
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[CLS]Binomial Theorem: Let $g$ and $h$ be complex numbers, and let $m_2$ be a natural number. In this case, $(h + g)^{m_2}$ can be represented as the summation $\sum_{m_1=0}^{m_2} \binom{m_2}{m_1} \times h^{-m_1 + m_2} \times g^{m_1}$.[SEP]
[CLS]Binomial Theorem: relates $ $ and $h$G complexeyYoung and let \ meg_2$ bs a naturalGamma. In this case, $(h + g)^{m_2)}(\ C be result as the summation $\}$._{m_1=0}^{m_2} \binom{m_2}{ My_1} \times oh}}/193_1 + bo_2})} \times g)^{-m])1}$.[SEP]
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[CLS]Integral of a Function: The integral of a integrable function $\int\limits_a^b f(z)\,\mathrm{d}z$ over an interval $\left(a,b\right)$ is equal to the difference of the antiderivative $U(b) - U(a)$.[SEP]
[CLS]Integral of a Function: The integral of a integrable function $\xt\limits_a^ba f(z)\,\mathrm{d}z$ Ai an interval $\left(a,b\right)$gi equal to the difference of the antiderivative $U(b) - U(a)$.[SEP]
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[CLS]Bayes law: Given two events $X$ and $C$, $P(X|C)$ can be computed using the formula $\mathbb{P}(C|X)\cdot \mathbb{P}(X)/\mathbb{P}(C) = \mathbb{P}(X|C)$, where $\mathbb{P}\left(X\right)$ and $\mathbb{P}(C)$ represent the probabilities of events $X$ and $C$ respectively, and $\mathbb{P}(C|X)$ represents the probability of event $C$ given event $X$.[SEP]
[CLS]Bayes law: Given two events $X$ and $C$, $P(X|C)$ can be computed using the formula $\mathbb{P}(C|X)\cdot \mathbb{P}(X)/\mathbb{P}(C) = \mathbb{P}(X|C)$, where $\mathbb{P}\left(X\right)$ and $\mathbb{P}(C)$ represent the probabilities of events $X$ and $C$ respectively, and $\mathbb{P}(C|X)$ represents the probability of event $C$ given event $X$.[SEP]
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[CLS]Integral of a Function: For any integrable function $u(y)$ and any real interval $\left(g,b\right)$ the integral over this interval $\int\limits_g^b u(y)\,\mathrm{d}y$ is equal to $-F(g) + F\left(b\right)$.[SEP]
[CLS]Integral of a Function: Forgi integrable function $u(y)$n any Ry interval $\left(g,b\right)$ the integral over t interval $\int)\,\limits_ (-^b u(y)\,\mathrm{d}y$ is equal to $-F(g) +ff\left(b\495)$.[SEP]
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[CLS]quadratic formula: Considering real numbers $r$ and $\vartheta$, the equation $z^2 + z \cdot r + \vartheta = 0$ has solutions given by $z_{1,2} = ((-1) r)/2 \pm \sqrt{r^2/4 - \vartheta}$.[SEP]
[CLS]quadadratic formula: Considering real)> -(r$ and $\ometry$,gt equation ((z^2 0 z \ dd r BG \vartheta = 0$ has solutions given byGdz_{1),_{-} { ((-1}}}{\ r)/2 -(pm \ TK{ ir_{-2/4 - \vartheta}$.[SEP]
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[CLS]Alternating Harmonic Series: For the alternating harmonic series $\sum_{h=1}^\infty \frac1h \cdot \left(-1\right)^{h + 1}$, the sum is equal to the natural logarithm of $2$, represented as $\sum_{h=1}^{\infty} \frac{(-1)^{h+1}}{h} = \ln(2)$.[SEP]
[CLS]Alternating mt First Series/( ff the alternatingdfrac series $\)}(}^{-h=1}^\inftyfgfrac1hggcdot |left(\1*right)^{h + 1}$, the sum is eq to the tangent Gi of $2752 represented GB $\sum_{h^(1})^{infty} \frac{(\(-014)^{h+1}}{h})} (\ \ln(2)$.[SEP]
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[CLS]Complex number summation: The sum of complex numbers $z$ and $w$ is $a + c + \left(d + b\right) \cdot j = z + w$, where $z = a + j\times b$ and $w = c + d \cdot j$.[SEP]
[CLS]Complex number summation: The sum of complex numbers $z$ and $w$ is $a + c + \left(d + mg\right) \cdot j = z + w$, where $z = a + j\times b$ and $w = c + d $-\cdot j$.[SEP]
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[CLS]Binomial coefficient formula: For all natural numbers $x$ and $k$ with $k \leq x$, we have that $\binom{x}{k} = \frac{1}{\left(-k + x\right)! k!}x!$.[SEP]
[CLS]Binomial coefficient formula: For all natural numbers $x$ and $k$ with $k \leq x$, we have that $\binom{x}{k} = \frac{1}{\left(-k + x\right)! k!}x!$.[SEP]
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[CLS]Third Binomial Formula: Considering real numbers $a$ and $h$, we can state that $(-h + a) \cdot \left(a + h\right)$ is equal to $a^2 - h^2$.[SEP]
[CLS]Third Binomialplication: Considering real numbers $a$ and $h$, Ve can state that $(-h + a) \cdot \left(a + h\right)$ is equal to $a^2 - h^2$.[SEP]
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[CLS]Definition of a factorial: 1*2*3*\dots*\chi = \chi![SEP]
[CLS] continuous of a factorial: 1*2*603*\ Distribution*\chi = \chi![SEP]
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[CLS]Sum of complex numbers: d_1 + e_1 \cdot \mathrm{i} + d_2 + \mathrm{i} \cdot e_2 = d_1 + d_2 + \mathrm{i} \cdot (e_1 + e_1)[SEP]
[CLS]Sum of common'): d_1 + e_}} \cdot \mathrm_{-i} + d_2 + \0018{i} \cdot e^{() = d_1 + d)*(-2 + \mathrm{ory} \cdot (e)*(-1 DG e_1)[SEP]
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[CLS]Poisson Distribution: \tfrac{\lambda^k}{k!} \cdot e^{-\lambda} = \mathbb{P}(R = k)[SEP]
[CLS]dpissonines: ($tfrac{\lambda^k}{k!} \cdot e^{-\lambda} = $-\}^{{P}(R = k)[SEP]
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[CLS]Eulers Identiy: \cos(\alpha) + j\sin(\alpha) = e^{j\alpha}[SEP]
[CLS]Eul Second IdentXY: \cos(\alpha), + j\sin(\alpha) = e^{j\alpha}[SEP]
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[CLS]Covariance: \operatorname{E}\left((R - \operatorname{E}\left(R\right))\cdot (-\operatorname{E}\left(Y\right) + Y)\right) = Cov\left(Y, R\right)[SEP]
[CLS]Covariance: \operatorname{E}\left((R - \operatorname{E}\left(R\right))\cdot (-\operatorname{E}\left(Y\right) + Y)\right) = Cov\left(Y, R\right)[SEP]
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[CLS]Binomial Formula: (a - c) \cdot (a - c) = c^2 + a^2 - 2 \cdot a \cdot c[SEP]
[CLS]Binomial Formula: (a - c)}\\cdot f(a - c) = c^2 + a^2 - 2 \cdot a \cdot c[SEP]
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[CLS]Power series definition for cosine function: The function $\cos{x}$ for any real number $x$ can be represented as the infinite series $\sum_{n=0}^\infty \frac{1}{x^{2n}*(2n)!}\left(-1\right)^n$.[SEP]
[CLS]Power series definition for cosine function: The function $\cos}{(x}$ for any real number $x956 can be represented as the infinite series $\sum_{\n=0}^\infty \frac{1}{x^{2n}}\*(642n)!}\left(-1\Ye)^n$.[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $a, b, c$, and opposite angle $\beta$ of $b$, it holds: $a^2 = b^2 + c^2 - \cos{\beta} c b*2$[SEP]
[CLS]ldf Cosines: In a triangle with $|F lengths $a, bations c$, another opposite sz ($beta$ of glbc955 ittg: $a^2 = b^2 $(- c^2 - \cos{\022} c BX*{-]$,[SEP]
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[CLS]De Morgan's Second Law (Set Version): The complement of the intersection of two sets $A \cap X$ is equal to the union of their complements $\overline{A} \cup \overline{X}$: $A \cap X^c = A^c \cup X^c$.[SEP]
[CLS]De Morgan's Second Law (Set Version): The complement of the intersection of two sets $A \cap X$ is equal to the union of their complements $\overline{A} \cup \overline{X}$: $A \cap X^c = A^c \cup X^c$.[SEP]
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[CLS]Sine Function Definition: \sin{y} = -y^7/7! + y - \frac{y^3}{3!} + \frac{1}{5!} y^5 \pm \dots[SEP]
[CLS]Sine Function Definition: \sin{y} = -y^7/7! + y - \frac{y^3}{3!} + \frac{1}{5!} y^5 \pm \dots[SEP]
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[CLS]Young's Inequality: r > 1, s > 1, \dfrac1r + 1/s = 1, 0 \leq h, b \geq 0 \Rightarrow h\cdot b \leq b^s/s + \frac{h^r}{r}[SEP]
[CLS]Young's Inequjt: r > 1, s > (/( \dfrac1r + measures/s = 1, 0 gyleqht, b \geq 0 \rmh\De ba \leq bg^s))/s = $(\frac{h'\r}{r)})[SEP]
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[CLS]Bayes probability rule: Given two events $D$ and $B$, the conditional probability $\mathbb{P}(D|B)$ can be computed using the formula $\mathbb{P}(D|B) = \mathbb{P}(B|D)\cdot \mathbb{P}(D)/\mathbb{P}(B)$, where $\mathbb{P}(D)$ and $P(B)$ represent the probabilities of events $D$ and $B$ respectively, and $P(B|D)$ represents the probability of event $B$ given event $D$.[SEP]
[CLS]Bayes probability rule: Given two events $D$ and $B$, the conditional probability $\mathbb{P}(D|B)$ can be computed using the formula $\mathbb{P}(D|B) = \mathbb{P}(B|D)\cdot \mathbb{P}(D)/\mathbb{P}(B)$, where $\mathbb{P}(D)$ and $P(B)$ represent the probabilities of events $D$ and $B$ respectively, and $P(B|D)$ represents the probability of event $B$ given event $D$.[SEP]
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[CLS]Determinant of 2x2 Matrix: det\left(\begin{array}{cc}g_1 & d\\f & g_2\end{array}\right) = g_2 g_1 - df[SEP]
[CLS]Determinant of 2xB2 Matrix\|_{ det^\}})\is484}{-array_{cc} (_1 & Rd\\ fn & rg_2\FI{}}{}\right) ( g_2 g_1 - df[SEP]
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[CLS]Variance: \dfrac{1}{m}\cdot \sum_{l=1}^m (x_l - \bar{X})^2 = Var[X],\bar{X} = (\sum_{k=0}^m x_k)/m[SEP]
[CLS]Variance: \dfrac{1}{m}\cdot \sum_{l=1}^m (x_l - \bar{X})^2 = Var[X],\bar{X} = $[sum_{k=0}^m x_k)/m[SEP]
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[CLS]Inverse of a 2x2 Matrix: Let $\begin{pmatrix}a & b\\c & d\end{pmatrix} = A$ be a 2x2 Matrix. Then we have $\frac1A = \frac{1}{\left|{A}\right|} \cdot [[d , -b],[-c , a]]$.[SEP]
[CLS]Inverse of ..., ...xF2 mt]], Let $\begin^{290})$a & b),\c & disend})}pmatrix} = Ra$ be a -x782 Matrix. trials we have $\frac951A = \fc{1}{\left|{A}{\right|} \ldots [[ di , ob)^\[-c , a]]$)[SEP]
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[CLS]Eulers Formula for polyhedra: If we consider a polyhedron with number of vertices $p$, edges $C$, and faces $D$, then the relationship $2 = p - C + D$ holds.[SEP]
[CLS]ot polyers Formula for polyvd derivatives: If we consider a roothedron with number of vertices $p$, edges $C$, and faces $ d$, then the relationship $2 = p - cl + D$ holdsxe[SEP]
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[CLS]Bayes probability rule: When considering two events $x$ and $F$, $\mathbb{P}(x|F)$ can be determined using the formula $\mathbb{P}(x|F) = \mathbb{P}(F|x)\cdot \mathbb{P}(x)/\mathbb{P}(F)$.[SEP]
[CLS]Bayes probability rule: When considering two events $x$ and $F$, $\mathbb{P}(x|F)$ can be determined using the formula $\mathbb{P}(x|F) = \mathbb{P}(F|x)\cdot \mathbb{P}(x)/\mathbb{P}(F)$.[SEP]
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[CLS]Integration by Parts: -\int\limits_a^d u(z) \cdot \frac{d}{dz} f(z)\,dz + f(d) \cdot u(d) - f(a) \cdot u(a) = \int\limits_a^d d/dz u(z) \cdot f(z)\,dz[SEP]
[CLS]Integrdefinition by planar: -\int)]60_{^{(\^d u\|_z) =cdot \frac}^{\d}{dz} f}[\z)}\dz GCtf(d) \cdot u(d)! - f(a) |\cdot u(}^{) (( \int\165_a^ Di d/dz g)/( dz) ogcdot f(z)\,dz[SEP]
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