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

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.

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

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 $(Y_1, Y_2)$ .

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.

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.

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)

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.