Random Variables & Distributions

Discrete Random Variables

• Takes countable values with assigned probabilities.
• Probability distribution: $P(X = x)$ for each value; $\sum P(x_i) = 1$.

Expected Value and Variance

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

Linear Transformations

• $E(aX + b) = aE(X) + b$
• $\text{Var}(aX + b) = a^2 \text{Var}(X)$

Combining Random Variables

• $E(X \pm Y) = E(X) \pm E(Y)$
• If independent: $\text{Var}(X \pm Y) = \text{Var}(X) + \text{Var}(Y)$ (always add!)

Binomial Distribution

$X \sim \text{Binomial}(n, p)$: number of successes in $n$ independent trials. $P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$ $\mu = np, \quad \sigma = \sqrt{np(1-p)}$

Geometric Distribution

$X \sim \text{Geometric}(p)$: number of trials until first success. $P(X = k) = (1-p)^{k-1}p$ $\mu = 1/p, \quad \sigma = \sqrt{(1-p)/p^2}$

Normal Distribution

$X \sim N(\mu, \sigma)$.

z-score: $z = \frac{x - \mu}{\sigma}$

68-95-99.7 Rule:

• 68% within 1σ, 95% within 2σ, 99.7% within 3σ.

Problem: A fair coin is flipped 10 times. Find $P(X = 7)$ where $X$ is the number of heads.

Solution: $P(X = 7) = \binom{10}{7}(0.5)^7(0.5)^3 = 120 \cdot (0.5)^{10} = 120/1024 \approx 0.117$

• Expected value: weighted average of outcomes.
• Variance of independent sums: always add variances.
• Binomial: fixed $n$ trials, binary outcomes, $\mu = np$.
• Geometric: trials until first success, $\mu = 1/p$.
• Normal: use z-scores and the 68-95-99.7 rule.