Probability Theory

Probability Theory Practice 1

Comprehensive practice problems covering events, distributions, expectations, and limit theorems

8 Problems

Suggested: 2 hours

Instructions

• Try to solve each problem before viewing the solution
• Click "Show Solution" to reveal the answer and detailed explanation
• Focus on understanding the problem-solving methodology

1Mutually Exclusive and Independent Events

Problem

Events A and B are mutually exclusive, B and C are independent, and $A \subset C$ .

Given: $P(\bar{A}C) = P(BC) = 0.2$ and $P(A \cup (B\bar{C})) = 0.5$

Find $P(A), P(B), P(C)$ .

Answer Summary

Use the event relations A ⊂ C, A ∩ B = ∅, and independence of B and C to turn the givens into a small probability system for P(A), P(B), and P(C).

2DNA Sequence Transition Probabilities

Problem

A DNA sequence consists of bases A, C, G, T. The transition probabilities from the ending base to the starting base of the next sequence are given in the following table:

End \ Start	A	G	C	T
A	0.1	0.2	0.3	0.4
G	0.2	0.2	0.2	0.3
C	0.3	0.2	0.2	0.1
T	0.4	0.3	0.3	0.2

(1) If sequences end with A, C, G, T with equal probability, find the probability that a sequence starts with each base.

(2) Under the assumption of (1), given a sequence starts with A, find the probability that the previous sequence ended with C or T.

Answer Summary

First average the transition probabilities with the law of total probability to get start-base frequencies, then reverse the direction with Bayes' theorem.

3Geometric Distribution and Degenerate Random Variables

Problem

(1) X and Y are independent, both following a geometric distribution with parameter p. Find $P(X = i | X + Y = n)$ for $i = 1, 2, \ldots, n-1$ .

(2) X is a degenerate random variable with distribution function F(x). Calculate:

\int_{-\infty}^{\infty} (F(x+a) - F(x))^{2026} dx

Answer Summary

Conditioning on X + Y = n makes every split i equally likely, while the degenerate CDF integral reduces to the interval where the step difference equals 1.

4Bivariate Density and Transformations

Problem

$X_1, X_2$ have joint density function:

f(x_1, x_2) = \begin{cases} \frac{1}{\sqrt{2\pi}} e^{-\frac{x_1^2 + x_2^2}{2}} & x_1 \in \mathbb{R}, 0 < x_2 < 1 \\ 0 & \text{otherwise} \end{cases}

ξ is independent of $(X_1, X_2)$ .

(1) If $Z = \xi X_1 + X_2$ , find the density function of Z.

(2) If $Y_1 = X_1/X_2, Y_2 = X_1 + X_2$ , find the joint density of $(Y_1, Y_2)$ .

Answer Summary

This problem tests marginal-density reasoning for linear combinations and Jacobian techniques for multivariable changes of variables.

5Expected Value with Sampling

Problem

There are 35 trees. 15 monkeys randomly choose trees to inhabit (each monkey independently and uniformly chooses one tree).

Randomly select 7 trees. Find the expected number of monkeys on these 7 trees.

Answer Summary

Introduce an indicator for each monkey landing in the chosen trees and apply linearity of expectation with success probability 7/35.

6Covariance of Monotone Functions and Bivariate Normal

Problem

(1) X and Y are independent. f(X) and g(X) are both monotonically non-decreasing (convex) Borel functions. Determine the sign of Cov(f(X), g(X)) and explain.

(2) Given $(X, Y) \sim N(2, 1, 4, 4, \frac{3}{4})$ , find the correlation coefficient of 2X + Y and 2X - Y.

Answer Summary

Use positive association for monotone functions of the same variable, then apply covariance and variance formulas to linear combinations of a bivariate normal vector.

7Central Limit Theorem Application: Porcelain Production

Problem

Statistics show that approximately 300 pieces of porcelain from a certain dynasty have survived to the present day.

The probability that any individual piece survives is approximately 0.005.

Calculate how many pieces needed to be produced so that there is an 80% probability that at least 300 pieces survive. (Given: Φ(0.84) = 0.8)

Answer Summary

Compute the mean and variance of the total defect count, then standardize the sum and apply the central limit theorem for an approximate probability.

8Coupon Collector Problem

Problem

A bubble tea brand launches a promotion: each purchase gives you one card. There are n different types of cards (n is a large integer).

Use the Law of Large Numbers to explain how many cups of bubble tea you need to buy to collect all n different types of cards.

Answer Summary

Break the collection process into stages, where each new coupon arrival has a geometric waiting time, and sum those expectations.

Related Theory

Review Probability Theory Fundamentals

Revisit the theory that supports these worked practice problems.

Review Random Variables and Distributions

Revisit the theory that supports these worked practice problems.

Review Numerical Characteristics of Random Variables

Revisit the theory that supports these worked practice problems.

Review Random Variable Limit Theorems

Revisit the theory that supports these worked practice problems.