Section 1: Problems

University-level exam questions for Probability Foundations.

Probability Basics

Problem 1.1

A medical test has sensitivity 0.95 (true positive rate) and specificity 0.90 (true negative rate). If the prevalence of the disease is 1%, what is the probability that a patient who tests positive actually has the disease?

Difficulty: Medium

Problem 1.2

Prove that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ from the axioms of probability.

Difficulty: Easy

Problem 1.3

Three cards are drawn from a standard 52-card deck without replacement. Find the probability that all three are aces.

Difficulty: Easy

Random Variables

Problem 2.1

Let $X$ be a continuous random variable with PDF $f(x) = 3x^2$ for $0 \le x \le 1$. Find the CDF, $P(X > 0.5)$, and the median.

Difficulty: Medium

Problem 2.2

If $X \sim \text{Uniform}(0, 1)$, find the PDF of $Y = -\ln(X)$. What distribution does $Y$ follow?

Difficulty: Medium

Common Distributions

Problem 3.1

Show that the Poisson distribution with parameter $\lambda$ can be derived as a limit of $\text{Binomial}(n, p)$ as $n \to \infty$ and $p \to 0$ with $np = \lambda$.

Difficulty: Hard

Problem 3.2

For a Gaussian random variable $X \sim \mathcal{N}(\mu, \sigma^2)$, derive the moment generating function $M_X(t) = E[e^{tX}]$.

Difficulty: Hard

Expectation and Variance

Problem 4.1

Let $X_1, \ldots, X_n$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Find $E[\bar{X}]$ and $\text{Var}(\bar{X})$ where $\bar{X} = \frac{1}{n}\sum_i X_i$.

Difficulty: Easy

Problem 4.2

Prove the law of total expectation: $E[X] = E[E[X \mid Y]]$.

Difficulty: Hard

Problem 4.3

Show that $\text{Var}(X) = E[X^2] - (E[X])^2$ using the definition of variance.

Difficulty: Easy

Challenge Problems

Problem 5.1

(Coupon collector) There are $n$ distinct types of coupons. Each time you collect one uniformly at random. Find the expected number of coupons you need to collect to have at least one of each type.

Difficulty: Very Hard

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Solutions

Solutions are available in the implementation file with verification code.