Add `Expectation Notebook`

This notebook introduces the concept of expectation in probability, detailing its definition, properties, and applications. It includes interactive visualizations for various probability distributions, calculations for expected values, and demonstrations of key properties such as linearity and the Law of the Unconscious Statistician (LOTUS). The content is structured to enhance understanding of expectation's role in decision-making under uncertainty.

probability/11_expectation.py

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+# /// script
+# requires-python = ">=3.10"
+# dependencies = [
+#     "marimo",
+#     "matplotlib==3.10.0",
+#     "numpy==2.2.3",
+#     "scipy==1.15.2",
+# ]
+# ///
+
+import marimo
+
+__generated_with = "0.11.19"
+app = marimo.App(width="medium", app_title="Expectation")
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        # Expectation
+
+        _This notebook is a computational companion to ["Probability for Computer Scientists"](https://chrispiech.github.io/probabilityForComputerScientists/en/part2/expectation/), by Stanford professor Chris Piech._
+
+        A random variable is fully represented by its Probability Mass Function (PMF), which describes each value the random variable can take on and the corresponding probabilities. However, a PMF can contain a lot of information. Sometimes it's useful to summarize a random variable with a single value!
+
+        The most common, and arguably the most useful, summary of a random variable is its **Expectation** (also called the expected value or mean).
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Definition of Expectation
+
+        The expectation of a random variable $X$, written $E[X]$, is the average of all the values the random variable can take on, each weighted by the probability that the random variable will take on that value.
+
+        $$E[X] = \sum_x x \cdot P(X=x)$$
+
+        Expectation goes by many other names: Mean, Weighted Average, Center of Mass, 1st Moment. All of these are calculated using the same formula.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Intuition Behind Expectation
+
+        The expected value represents the long-run average value of a random variable over many independent repetitions of an experiment.
+
+        For example, if you roll a fair six-sided die many times and calculate the average of all rolls, that average will approach the expected value of 3.5 as the number of rolls increases.
+
+        Let's visualize this concept:
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(np, plt):
+    # Set random seed for reproducibility
+    np.random.seed(42)
+
+    # Simulate rolling a die many times
+    exp_num_rolls = 1000
+    exp_die_rolls = np.random.randint(1, 7, size=exp_num_rolls)
+
+    # Calculate the running average
+    exp_running_avg = np.cumsum(exp_die_rolls) / np.arange(1, exp_num_rolls + 1)
+
+    # Create the plot
+    plt.figure(figsize=(10, 5))
+    plt.plot(range(1, exp_num_rolls + 1), exp_running_avg, label='Running Average')
+    plt.axhline(y=3.5, color='r', linestyle='--', label='Expected Value (3.5)')
+    plt.xlabel('Number of Rolls')
+    plt.ylabel('Average Value')
+    plt.title('Running Average of Die Rolls Approaching Expected Value')
+    plt.legend()
+    plt.grid(alpha=0.3)
+    plt.xscale('log')  # Log scale to better see convergence
+
+    # Add annotations
+    plt.annotate('As the number of rolls increases,\nthe average approaches the expected value',
+                xy=(exp_num_rolls, exp_running_avg[-1]), xytext=(exp_num_rolls/3, 4),
+                arrowprops=dict(facecolor='black', shrink=0.05, width=1.5))
+
+    plt.gca()  # Return the current axes to ensure proper display
+    return exp_die_rolls, exp_num_rolls, exp_running_avg
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(r"""## Properties of Expectation""")
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.accordion(
+        {
+            "1. Linearity of Expectation": mo.md(
+                r"""
+                $$E[aX + b] = a \cdot E[X] + b$$
+
+                Where $a$ and $b$ are constants (not random variables).
+
+                This means that if you multiply a random variable by a constant, the expectation is multiplied by that constant. And if you add a constant to a random variable, the expectation increases by that constant.
+                """
+            ),
+            "2. Expectation of the Sum of Random Variables": mo.md(
+                r"""
+                $$E[X + Y] = E[X] + E[Y]$$
+
+                This is true regardless of the relationship between $X$ and $Y$. They can be dependent, and they can have different distributions. This also applies with more than two random variables:
+
+                $$E\left[\sum_{i=1}^n X_i\right] = \sum_{i=1}^n E[X_i]$$
+                """
+            ),
+            "3. Law of the Unconscious Statistician (LOTUS)": mo.md(
+                r"""
+                $$E[g(X)] = \sum_x g(x) \cdot P(X=x)$$
+
+                This allows us to calculate the expected value of a function $g(X)$ of a random variable $X$ when we know the probability distribution of $X$ but don't explicitly know the distribution of $g(X)$.
+
+                This theorem has the humorous name "Law of the Unconscious Statistician" (LOTUS) because it's so useful that you should be able to employ it unconsciously.
+                """
+            ),
+            "4. Expectation of a Constant": mo.md(
+                r"""
+                $$E[a] = a$$
+
+                Sometimes in proofs, you'll end up with the expectation of a constant (rather than a random variable). Since a constant doesn't change, its expected value is just the constant itself.
+                """
+            ),
+        }
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Calculating Expectation
+
+        Let's calculate the expected value for some common examples:
+
+        ### Example 1: Fair Die Roll
+
+        For a fair six-sided die, the PMF is:
+
+        $$P(X=x) = \frac{1}{6} \text{ for } x \in \{1, 2, 3, 4, 5, 6\}$$
+
+        The expected value is:
+
+        $$E[X] = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6} = \frac{21}{6} = 3.5$$
+
+        Let's implement this calculation in Python:
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    def calc_expectation_die():
+        """Calculate the expected value of a fair six-sided die roll."""
+        exp_die_values = range(1, 7)
+        exp_die_probs = [1/6] * 6
+
+        exp_die_expected = sum(x * p for x, p in zip(exp_die_values, exp_die_probs))
+        return exp_die_expected
+
+    exp_die_result = calc_expectation_die()
+    print(f"Expected value of a fair die roll: {exp_die_result}")
+    return calc_expectation_die, exp_die_result
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ### Example 2: Sum of Two Dice
+
+        Now let's calculate the expected value for the sum of two fair dice. First, we need the PMF:
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    def pmf_sum_two_dice(y_val):
+        """Returns the probability that the sum of two dice is y."""
+        # Count the number of ways to get sum y
+        exp_count = 0
+        for dice1 in range(1, 7):
+            for dice2 in range(1, 7):
+                if dice1 + dice2 == y_val:
+                    exp_count += 1
+        return exp_count / 36  # There are 36 possible outcomes (6×6)
+
+    # Test the function for a few values
+    exp_test_values = [2, 7, 12]
+    for exp_test_y in exp_test_values:
+        print(f"P(Y = {exp_test_y}) = {pmf_sum_two_dice(exp_test_y)}")
+    return exp_test_values, exp_test_y, pmf_sum_two_dice
+
+
+@app.cell
+def _(pmf_sum_two_dice):
+    def calc_expectation_sum_two_dice():
+        """Calculate the expected value of the sum of two dice."""
+        exp_sum_two_dice = 0
+        # Sum of dice can take on the values 2 through 12
+        for exp_x in range(2, 13):
+            exp_pr_x = pmf_sum_two_dice(exp_x)  # PMF gives P(sum is x)
+            exp_sum_two_dice += exp_x * exp_pr_x
+        return exp_sum_two_dice
+
+    exp_sum_result = calc_expectation_sum_two_dice()
+
+    # Round to 2 decimal places for display
+    exp_sum_result_rounded = round(exp_sum_result, 2)
+
+    print(f"Expected value of the sum of two dice: {exp_sum_result_rounded}")
+
+    # Let's also verify this with a direct calculation
+    exp_direct_calc = sum(x * pmf_sum_two_dice(x) for x in range(2, 13))
+    exp_direct_calc_rounded = round(exp_direct_calc, 2)
+
+    print(f"Direct calculation: {exp_direct_calc_rounded}")
+
+    # Verify that this equals 7
+    print(f"Is the expected value exactly 7? {abs(exp_sum_result - 7) < 1e-10}")
+    return (
+        calc_expectation_sum_two_dice,
+        exp_direct_calc,
+        exp_direct_calc_rounded,
+        exp_sum_result,
+        exp_sum_result_rounded,
+    )
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ### Visualizing Expectation
+
+        Let's visualize the expectation for the sum of two dice. The expected value is the "center of mass" of the PMF:
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(plt, pmf_sum_two_dice):
+    # Create the visualization
+    exp_y_values = list(range(2, 13))
+    exp_probabilities = [pmf_sum_two_dice(y) for y in exp_y_values]
+
+    dice_fig, dice_ax = plt.subplots(figsize=(10, 5))
+    dice_ax.bar(exp_y_values, exp_probabilities, width=0.4)
+    dice_ax.axvline(x=7, color='r', linestyle='--', linewidth=2, label='Expected Value (7)')
+
+    dice_ax.set_xticks(exp_y_values)
+    dice_ax.set_xlabel('Sum of two dice (y)')
+    dice_ax.set_ylabel('Probability: P(Y = y)')
+    dice_ax.set_title('PMF of Sum of Two Dice with Expected Value')
+    dice_ax.grid(alpha=0.3)
+    dice_ax.legend()
+
+    # Add probability values on top of bars
+    for exp_i, exp_prob in enumerate(exp_probabilities):
+        dice_ax.text(exp_y_values[exp_i], exp_prob + 0.001, f'{exp_prob:.3f}', ha='center')
+
+    plt.tight_layout()
+    plt.gca()
+    return dice_ax, dice_fig, exp_i, exp_prob, exp_probabilities, exp_y_values
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Demonstrating the Properties of Expectation
+
+        Let's demonstrate some of these properties with examples:
+        """
+    )
+    return
+
+
+@app.cell
+def _(exp_die_result):
+    # Demonstrate linearity of expectation (1)
+    # E[aX + b] = a*E[X] + b
+
+    # For a die roll X with E[X] = 3.5
+    prop_a = 2
+    prop_b = 10
+
+    # Calculate E[2X + 10] using the property
+    prop_expected_using_property = prop_a * exp_die_result + prop_b
+    prop_expected_using_property_rounded = round(prop_expected_using_property, 2)
+
+    print(f"Using linearity property: E[{prop_a}X + {prop_b}] = {prop_a} * E[X] + {prop_b} = {prop_expected_using_property_rounded}")
+
+    # Calculate E[2X + 10] directly
+    prop_expected_direct = sum((prop_a * x + prop_b) * (1/6) for x in range(1, 7))
+    prop_expected_direct_rounded = round(prop_expected_direct, 2)
+
+    print(f"Direct calculation: E[{prop_a}X + {prop_b}] = {prop_expected_direct_rounded}")
+
+    # Verify they match
+    print(f"Do they match? {abs(prop_expected_using_property - prop_expected_direct) < 1e-10}")
+    return (
+        prop_a,
+        prop_b,
+        prop_expected_direct,
+        prop_expected_direct_rounded,
+        prop_expected_using_property,
+        prop_expected_using_property_rounded,
+    )
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ### Law of the Unconscious Statistician (LOTUS)
+
+        Let's use LOTUS to calculate $E[X^2]$ for a die roll, which will be useful when we study variance:
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    # Calculate E[X^2] for a die roll using LOTUS (3)
+    lotus_die_values = range(1, 7)
+    lotus_die_probs = [1/6] * 6
+
+    # Using LOTUS: E[X^2] = sum(x^2 * P(X=x))
+    lotus_expected_x_squared = sum(x**2 * p for x, p in zip(lotus_die_values, lotus_die_probs))
+    lotus_expected_x_squared_rounded = round(lotus_expected_x_squared, 2)
+
+    expected_x_squared = 3.5**2
+    expected_x_squared_rounded = round(expected_x_squared, 2)
+
+    print(f"E[X^2] for a die roll = {lotus_expected_x_squared_rounded}")
+    print(f"(E[X])^2 for a die roll = {expected_x_squared_rounded}")
+    return (
+        expected_x_squared,
+        expected_x_squared_rounded,
+        lotus_die_probs,
+        lotus_die_values,
+        lotus_expected_x_squared,
+        lotus_expected_x_squared_rounded,
+    )
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        /// Note
+        Note that E[X^2] != (E[X])^2
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Interactive Example
+
+        Let's explore how the expected value changes as we adjust the parameters of common probability distributions. This interactive visualization focuses specifically on the relationship between distribution parameters and expected values.
+
+        Use the controls below to select a distribution and adjust its parameters. The graph will show how the expected value changes across a range of parameter values.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    # Create UI elements for distribution selection
+    dist_selection = mo.ui.dropdown(
+        options=[
+            "bernoulli",
+            "binomial",
+            "geometric",
+            "poisson"
+        ],
+        value="bernoulli",
+        label="Select a distribution"
+    )
+    return (dist_selection,)
+
+
+@app.cell(hide_code=True)
+def _(dist_selection):
+    dist_selection.center()
+    return
+
+
+@app.cell(hide_code=True)
+def _(dist_description):
+    dist_description
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md("""### Adjust Parameters""")
+    return
+
+
+@app.cell(hide_code=True)
+def _(controls):
+    controls
+    return
+
+
+@app.cell(hide_code=True)
+def _(
+    dist_selection,
+    lambda_range,
+    np,
+    param_lambda,
+    param_n,
+    param_p,
+    param_range,
+    plt,
+):
+    # Calculate expected values based on the selected distribution
+    if dist_selection.value == "bernoulli":
+        # Get parameter range for visualization
+        p_min, p_max = param_range.value
+        param_values = np.linspace(p_min, p_max, 100)
+
+        # E[X] = p for Bernoulli
+        expected_values = param_values
+        current_param = param_p.value
+        current_expected = round(current_param, 2)
+        x_label = "p (probability of success)"
+        title = "Expected Value of Bernoulli Distribution"
+        formula = "E[X] = p"
+
+    elif dist_selection.value == "binomial":
+        # Get parameter range for visualization
+        p_min, p_max = param_range.value
+        param_values = np.linspace(p_min, p_max, 100)
+
+        # E[X] = np for Binomial
+        n = int(param_n.value)
+        expected_values = [n * p for p in param_values]
+        current_param = param_p.value
+        current_expected = round(n * current_param, 2)
+        x_label = "p (probability of success)"
+        title = f"Expected Value of Binomial Distribution (n={n})"
+        formula = f"E[X] = n × p = {n} × p"
+
+    elif dist_selection.value == "geometric":
+        # Get parameter range for visualization
+        p_min, p_max = param_range.value
+        # Ensure p is not 0 for geometric distribution
+        p_min = max(0.01, p_min)
+        param_values = np.linspace(p_min, p_max, 100)
+
+        # E[X] = 1/p for Geometric
+        expected_values = [1/p for p in param_values]
+        current_param = param_p.value
+        current_expected = round(1 / current_param, 2)
+        x_label = "p (probability of success)"
+        title = "Expected Value of Geometric Distribution"
+        formula = "E[X] = 1/p"
+
+    else:  # Poisson
+        # Get parameter range for visualization
+        lambda_min, lambda_max = lambda_range.value
+        param_values = np.linspace(lambda_min, lambda_max, 100)
+
+        # E[X] = lambda for Poisson
+        expected_values = param_values
+        current_param = param_lambda.value
+        current_expected = round(current_param, 2)
+        x_label = "λ (rate parameter)"
+        title = "Expected Value of Poisson Distribution"
+        formula = "E[X] = λ"
+
+    # Create the plot
+    dist_fig, dist_ax = plt.subplots(figsize=(10, 6))
+
+    # Plot the expected value function
+    dist_ax.plot(param_values, expected_values, 'b-', linewidth=2, label="Expected Value Function")
+
+    # Mark the current parameter value
+    dist_ax.plot(current_param, current_expected, 'ro', markersize=10, label=f"Current Value: E[X] = {current_expected}")
+
+    # Add a horizontal line from y-axis to the current point
+    dist_ax.hlines(current_expected, param_values[0], current_param, colors='r', linestyles='dashed')
+
+    # Add a vertical line from x-axis to the current point
+    dist_ax.vlines(current_param, 0, current_expected, colors='r', linestyles='dashed')
+
+    # Add shaded area under the curve
+    dist_ax.fill_between(param_values, 0, expected_values, alpha=0.2, color='blue')
+
+    dist_ax.set_xlabel(x_label, fontsize=12)
+    dist_ax.set_ylabel("Expected Value: E[X]", fontsize=12)
+    dist_ax.set_title(title, fontsize=14, fontweight='bold')
+    dist_ax.grid(True, alpha=0.3)
+
+    # Move legend to lower right to avoid overlap with formula
+    dist_ax.legend(loc='lower right', fontsize=10)
+
+    # Add formula text box in upper left
+    dist_props = dict(boxstyle='round', facecolor='white', alpha=0.8)
+    dist_ax.text(0.02, 0.95, formula, transform=dist_ax.transAxes, fontsize=12,
+            verticalalignment='top', bbox=dist_props)
+
+    # Set reasonable y-axis limits based on the distribution
+    if dist_selection.value == "geometric":
+        max_y = min(50, 2/max(0.01, param_values[0]))
+        dist_ax.set_ylim(0, max_y)
+    elif dist_selection.value == "binomial":
+        dist_ax.set_ylim(0, int(param_n.value) + 1)
+    else:
+        dist_ax.set_ylim(0, max(expected_values) * 1.1)
+
+    # Add annotations for current value
+    annotation_x = current_param + (param_values[-1] - param_values[0]) * 0.05
+    annotation_y = current_expected
+
+    # Adjust annotation position if it would go off the chart
+    if annotation_x > param_values[-1] * 0.9:
+        annotation_x = current_param - (param_values[-1] - param_values[0]) * 0.2
+
+    dist_ax.annotate(
+        f"Parameter: {current_param:.2f}\nE[X] = {current_expected}",
+        xy=(current_param, current_expected),
+        xytext=(annotation_x, annotation_y),
+        arrowprops=dict(facecolor='black', shrink=0.05, width=1.5, alpha=0.7),
+        bbox=dist_props
+    )
+
+    plt.tight_layout()
+    plt.gca()
+    return (
+        annotation_x,
+        annotation_y,
+        current_expected,
+        current_param,
+        dist_ax,
+        dist_fig,
+        dist_props,
+        expected_values,
+        formula,
+        lambda_max,
+        lambda_min,
+        max_y,
+        n,
+        p_max,
+        p_min,
+        param_values,
+        title,
+        x_label,
+    )
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Expectation vs. Mode
+
+        The expected value (mean) of a random variable is not always the same as its most likely value (mode). Let's explore this with an example:
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(np, plt, stats):
+    # Create a skewed distribution
+    skew_n = 10
+    skew_p = 0.25
+
+    # Binomial PMF
+    skew_x_values = np.arange(0, skew_n+1)
+    skew_pmf_values = stats.binom.pmf(skew_x_values, skew_n, skew_p)
+
+    # Find the mode (most likely value)
+    skew_mode = skew_x_values[np.argmax(skew_pmf_values)]
+
+    # Calculate the expected value
+    skew_expected = skew_n * skew_p
+    skew_expected_rounded = round(skew_expected, 2)
+
+    # Create the plot
+    skew_fig, skew_ax = plt.subplots(figsize=(10, 5))
+    skew_ax.bar(skew_x_values, skew_pmf_values, alpha=0.7, width=0.4)
+
+    # Add vertical lines for mode and expected value
+    skew_ax.axvline(x=skew_mode, color='g', linestyle='--', linewidth=2, 
+                label=f'Mode = {skew_mode} (Most likely value)')
+    skew_ax.axvline(x=skew_expected, color='r', linestyle='--', linewidth=2, 
+                label=f'Expected Value = {skew_expected_rounded} (Mean)')
+
+    # Add annotations to highlight the difference
+    skew_ax.annotate('Mode', xy=(skew_mode, 0.05), xytext=(skew_mode-2.0, 0.1),
+                arrowprops=dict(facecolor='green', shrink=0.05, width=1.5), color='green')
+    skew_ax.annotate('Expected Value', xy=(skew_expected, 0.05), xytext=(skew_expected+1, 0.15),
+                arrowprops=dict(facecolor='red', shrink=0.05, width=1.5), color='red')
+
+    # Highlight the difference between mode and expected value
+    if skew_mode != int(skew_expected):
+        # Add a shaded region between mode and expected value
+        min_x = min(skew_mode, skew_expected)
+        max_x = max(skew_mode, skew_expected)
+        skew_ax.axvspan(min_x, max_x, alpha=0.2, color='purple')
+
+        # Add text explaining the difference
+        mid_x = (skew_mode + skew_expected) / 2
+        skew_ax.text(mid_x, max(skew_pmf_values) * 0.5, 
+                 f"Difference: {abs(skew_mode - skew_expected_rounded):.2f}",
+                 ha='center', va='center', bbox=dict(facecolor='white', alpha=0.7))
+
+    skew_ax.set_xlabel('Number of Successes')
+    skew_ax.set_ylabel('Probability')
+    skew_ax.set_title(f'Binomial Distribution (n={skew_n}, p={skew_p})')
+    skew_ax.grid(alpha=0.3)
+    skew_ax.legend()
+
+    plt.tight_layout()
+    plt.gca()
+    return (
+        max_x,
+        mid_x,
+        min_x,
+        skew_ax,
+        skew_expected,
+        skew_expected_rounded,
+        skew_fig,
+        skew_mode,
+        skew_n,
+        skew_p,
+        skew_pmf_values,
+        skew_x_values,
+    )
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        /// NOTE
+        For the sum of two dice we calculated earlier, we found the expected value to be exactly 7. In that case, 7 also happens to be the mode (most likely outcome) of the distribution. However, this is just a coincidence for this particular example!
+
+        As we can see from the binomial distribution above, the expected value (2.50) and the mode (2) are often different values (this is common in skewed distributions). The expected value represents the "center of mass" of the distribution, while the mode represents the most likely single outcome.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Applications of Expectation
+
+        Expected value has numerous applications across various fields:
+
+        1. **Finance**: Expected return on investments, option pricing
+        2. **Insurance**: Setting premiums based on expected claims
+        3. **Gaming**: Calculating the expected winnings in games of chance
+        4. **Machine Learning**: Loss function minimization
+        5. **Operations Research**: Decision making under uncertainty
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## 🤔 Test Your Understanding
+
+        Choose what you believe are the correct options in the questions below:
+
+        <details>
+        <summary>The expected value of a random variable is always one of the possible values the random variable can take.</summary>
+        ❌ False! The expected value is a weighted average and may not be a value the random variable can actually take. For example, the expected value of a fair die roll is 3.5, which is not a possible outcome.
+        </details>
+
+        <details>
+        <summary>If X and Y are independent random variables, then E[X·Y] = E[X]·E[Y].</summary>
+        ✅ True! For independent random variables, the expectation of their product equals the product of their expectations.
+        </details>
+
+        <details>
+        <summary>The expected value of a constant random variable (one that always takes the same value) is that constant.</summary>
+        ✅ True! If X = c with probability 1, then E[X] = c.
+        </details>
+
+        <details>
+        <summary>The expected value of the sum of two random variables is always the sum of their expected values, regardless of whether they are independent.</summary>
+        ✅ True! This is the linearity of expectation property: E[X + Y] = E[X] + E[Y], which holds regardless of dependence.
+        </details>
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Practical Applications of Expectation
+
+        Expected values show up everywhere - from investment decisions and insurance pricing to machine learning algorithms and game design. Engineers use them to predict system reliability, data scientists to understand customer behavior, and economists to model market outcomes. They're essential for risk assessment in project management and for optimizing resource allocation in operations research.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Key Takeaways
+
+        Expectation gives us a single value that summarizes a random variable's central tendency - it's the weighted average of all possible outcomes, where the weights are probabilities. The linearity property makes expectations easy to work with, even for complex combinations of random variables. While a PMF gives the complete probability picture, expectation provides an essential summary that helps us make decisions under uncertainty. In our next notebook, we'll explore variance, which measures how spread out a random variable's values are around its expectation.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(r"""#### Appendix (containing helper code)""")
+    return
+
+
+@app.cell(hide_code=True)
+def _():
+    import marimo as mo
+    return (mo,)
+
+
+@app.cell(hide_code=True)
+def _():
+    import matplotlib.pyplot as plt
+    import numpy as np
+    from scipy import stats
+    import collections
+    return collections, np, plt, stats
+
+
+@app.cell(hide_code=True)
+def _(dist_selection, mo):
+    # Parameter controls for probability-based distributions
+    param_p = mo.ui.slider(
+        start=0.01, 
+        stop=0.99, 
+        step=0.01, 
+        value=0.5, 
+        label="p (probability of success)",
+        full_width=True
+    )
+
+    # Parameter control for binomial distribution
+    param_n = mo.ui.slider(
+        start=1, 
+        stop=50, 
+        step=1, 
+        value=10, 
+        label="n (number of trials)",
+        full_width=True
+    )
+
+    # Parameter control for Poisson distribution
+    param_lambda = mo.ui.slider(
+        start=0.1, 
+        stop=20, 
+        step=0.1, 
+        value=5, 
+        label="λ (rate parameter)",
+        full_width=True
+    )
+
+    # Parameter range sliders for visualization
+    param_range = mo.ui.range_slider(
+        start=0, 
+        stop=1, 
+        step=0.01, 
+        value=[0, 1], 
+        label="Parameter range to visualize",
+        full_width=True
+    )
+
+    lambda_range = mo.ui.range_slider(
+        start=0, 
+        stop=20, 
+        step=0.1, 
+        value=[0, 20], 
+        label="λ range to visualize",
+        full_width=True
+    )
+
+    # Display appropriate controls based on the selected distribution
+    if dist_selection.value == "bernoulli":
+        controls = mo.hstack([param_p, param_range], justify="space-around")
+    elif dist_selection.value == "binomial":
+        controls = mo.hstack([param_p, param_n, param_range], justify="space-around")
+    elif dist_selection.value == "geometric":
+        controls = mo.hstack([param_p, param_range], justify="space-around")
+    else:  # poisson
+        controls = mo.hstack([param_lambda, lambda_range], justify="space-around")
+    return controls, lambda_range, param_lambda, param_n, param_p, param_range
+
+
+@app.cell(hide_code=True)
+def _(dist_selection, mo):
+    # Create distribution descriptions based on selection
+    if dist_selection.value == "bernoulli":
+        dist_description = mo.md(
+            r"""
+            **Bernoulli Distribution**
+
+            A Bernoulli distribution models a single trial with two possible outcomes: success (1) or failure (0).
+
+            - Parameter: $p$ = probability of success
+            - Expected Value: $E[X] = p$
+            - Example: Flipping a coin once (p = 0.5 for a fair coin)
+            """
+        )
+    elif dist_selection.value == "binomial":
+        dist_description = mo.md(
+            r"""
+            **Binomial Distribution**
+
+            A Binomial distribution models the number of successes in $n$ independent trials.
+
+            - Parameters: $n$ = number of trials, $p$ = probability of success
+            - Expected Value: $E[X] = np$
+            - Example: Number of heads in 10 coin flips
+            """
+        )
+    elif dist_selection.value == "geometric":
+        dist_description = mo.md(
+            r"""
+            **Geometric Distribution**
+
+            A Geometric distribution models the number of trials until the first success.
+
+            - Parameter: $p$ = probability of success
+            - Expected Value: $E[X] = \frac{1}{p}$
+            - Example: Number of coin flips until first heads
+            """
+        )
+    else:  # poisson
+        dist_description = mo.md(
+            r"""
+            **Poisson Distribution**
+
+            A Poisson distribution models the number of events occurring in a fixed interval.
+
+            - Parameter: $\lambda$ = average rate of events
+            - Expected Value: $E[X] = \lambda$
+            - Example: Number of emails received per hour
+            """
+        )
+    return (dist_description,)
+
+
+if __name__ == "__main__":
+    app.run()