Mean and Standard Deviation of a Discrete Random Variable

🎲 Discrete Random Variables

Part 1 of 7 — Probability Distributions

What Is a Random Variable?

A random variable assigns a numerical value to each outcome of a random process.

🔑 Key Idea: A discrete random variable has a probability distribution that lists every possible value and its probability.

Probability Distribution Table

Requirements:

• Every probability is between 0 and 1: $0 \\leq P(X = x) \\leq 1$
• All probabilities sum to 1: $\\sum P(X = x) = 1$

Reading the Table

From the table above:

• $P(X = 2) = 0.4$
• $P(X \\geq 2) = P(X=2) + P(X=3) = 0.4 + 0.2 = 0.6$
• $P(X < 2) = P(X=0) + P(X=1) = 0.1 + 0.3 = 0.4$

Concept Check U0001f3af

Probability Distribution Practice 🧮

Given: $P(X=1) = 0.25$, $P(X=2) = 0.35$, $P(X=3) = 0.30$, $P(X=4) = 0.10$

1) $P(X \\leq 2)$?

2) $P(X > 2)$?

3) Do the probabilities sum to 1? (yes or no)

🎯 Expected Value (Mean of a Random Variable)

Part 2 of 7 — Expected Value

The Mean of a Discrete Random Variable

$\\mu_X = E(X) = \\sum x_i \\cdot P(x_i)$

The expected value is the long-run average — if you repeated the random process many times, the average outcome would approach $E(X)$.

Worked Example

$E(X) = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2) = 0 + 0.3 + 0.8 + 0.6 = 1.7$

⚠️ The expected value does NOT have to be a possible outcome. $X$ can’t actually equal 1.7, but 1.7 is the long-run average.

Interpretation on the AP Exam

“If we were to repeat this random process many, many times, the average value of $X$ would be approximately 1.7.”

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Expected Value Calculation 🧮

A game costs $5 to play. You win $20 with probability 0.2, win $5 with probability 0.3, and win $0 with probability 0.5.

Let $X$ = net gain (winnings minus cost).

1) $X$ values: $15, $0, and $\\underline{\\quad}$ (third net gain value)

2) $E(X) = ?$ (expected net gain)

3) Is this game favorable for the player? (yes or no)

📊 Variance & Standard Deviation of a Random Variable

Part 3 of 7 — Spread of a Distribution

Variance

$\\text{Var}(X) = \\sigma_X^2 = \\sum (x_i - \\mu_X)^2 \\cdot P(x_i)$

Standard Deviation

$\\sigma_X = \\sqrt{\\text{Var}(X)}$

Worked Example

Using our distribution: $\\mu_X = 1.7$

$\\sigma^2 = 0.289 + 0.147 + 0.036 + 0.338 = 0.81$ $\\sigma = \\sqrt{0.81} = 0.9$

🔑 Standard deviation measures the typical distance of outcomes from the mean.

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Variance Drill 🧮

A random variable $X$ has: $\\mu = 3$, and outcomes $X = 1, 3, 5$ each with probability $1/3$.

1) $(1 - 3)^2 \\cdot (1/3) = ?$ (first term of variance calculation)

2) $(3 - 3)^2 \\cdot (1/3) = ?$

3) $\\sigma^2 = ?$ (total variance, as a fraction like a/b)

⚖️ Transforming Random Variables

Part 4 of 7 — Linear Transformations

Rules for $Y = a + bX$

🔑 Adding a constant shifts the center but does NOT change spread. Multiplying by a constant scales both center and spread.

Example: Temperature Conversion

If $X$ is temperature in Celsius with $\\mu_X = 20$ and $\\sigma_X = 5$:

$Y = 32 + 1.8X$ (Fahrenheit)

• $\\mu_Y = 32 + 1.8(20) = 68$°F
• $\\sigma_Y = 1.8(5) = 9$°F

The mean shifts AND scales; the standard deviation only scales (adding 32 has no effect on spread).

Concept Check U0001f3af

Transformation Practice 🧮

$X$ has $\\mu_X = 50$ and $\\sigma_X = 10$. Let $Y = 2X - 30$.

1) $\\mu_Y = ?$

2) $\\sigma_Y = ?$

3) $\\sigma_Y^2 = ?$ (variance of $Y$)

🔄 Combining Independent Random Variables

Part 5 of 7 — Sums & Differences

Rules for Independent Random Variables

If $X$ and $Y$ are independent:

⚠️ Critical: Variances always ADD, even for differences! Standard deviations do NOT simply add or subtract.

Why Variances Add for Differences

Think of it this way: whether you add or subtract, the uncertainty (variability) in each variable contributes to the total uncertainty. Subtracting doesn’t reduce uncertainty — it compounds it.

Example

$X$: exam score, $\\mu_X = 75$, $\\sigma_X = 8$ $Y$: quiz score, $\\mu_Y = 85$, $\\sigma_Y = 6$

$X + Y$: $\\mu = 160$, $\\sigma^2 = 64 + 36 = 100$, $\\sigma = 10$ $X - Y$: $\\mu = -10$, $\\sigma^2 = 64 + 36 = 100$, $\\sigma = 10$

Concept Check U0001f3af

Combining Variables 🧮

$X$: $\\mu_X = 100$, $\\sigma_X = 12$. $Y$: $\\mu_Y = 80$, $\\sigma_Y = 5$. Independent.

1) $E(X + Y) = ?$

2) $\\text{Var}(X - Y) = ?$

3) $\\sigma_{X+Y} = ?$ (round to 1 decimal)

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Problems

Strategy for Random Variable Problems

1. Identify the random variable and its distribution
2. Calculate $E(X)$ using $\\sum x \\cdot P(x)$
3. Apply transformation rules if $Y = a + bX$
4. Combine using variance addition for independent variables
5. Interpret in context for full AP credit

AP Exam Tip

When asked to interpret expected value: “If [process] were repeated many times, the average [variable] would be approximately [value].”

When asked to interpret standard deviation: “The [variable] typically varies by about [value] from the mean of [mean].”

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Insurance Problem 🧮

An insurance company charges $300/year. Claims: $0 (prob 0.9), $1000 (prob 0.08), $5000 (prob 0.02).

1) Expected claim per customer?

2) Expected profit per customer?

3) Standard deviation of claims? (round to nearest dollar)

📝 Review & Applications

Part 7 of 7 — Comprehensive Review

Key Formulas Summary

Common Mistakes on the AP Exam

• Adding standard deviations instead of variances
• Subtracting variances for $X - Y$
• Forgetting that $E(X)$ doesn’t have to be a possible value
• Confusing “expected value” with “most likely value”

Concept Check U0001f3af

Final Challenge 🧮

$X$: $\\mu = 20$, $\\sigma = 3$. $Y$: $\\mu = 15$, $\\sigma = 4$. Independent.

1) $E(2X + Y) = ?$

2) $\\text{Var}(2X + Y) = ?$

3) $\\sigma_{2X+Y} = ?$ (round to 2 decimals)