Probability Theory Practice 2

Given events A, B, C where $P(A) = P(B) = P(C) = \frac{1}{4}$, $P(AB) = P(BC) = 0$, and $P(AC) = \frac{1}{8}$.

Find the probability that at least one of A, B, C occurs.

Determine which of the following are valid characteristic functions:

• $\varphi_1(t) = \cos(t)$
• $\varphi_2(t) = \cos^2(t)$
• $\varphi_3(t) = |\cos(t)|$
• $\varphi_4(t) = \cos(|t|)$

Random variables X and Y have joint density:

(1) Find the constant c.

(2) Find the marginal densities $f_X(x)$ and $f_Y(y)$.

(3) Are X and Y independent?

(4) Find $P(X + Y < 1)$.

Let X have moment generating function $M_X(t) = \frac{1}{1-2t}$ for $t < \frac{1}{2}$.

(1) Identify the distribution of X.

(2) Find $E[X]$ and $\text{Var}(X)$.

(3) If $X_1, X_2, \ldots, X_n$ are i.i.d. copies of X, find the MGF of $S_n = X_1 + X_2 + \cdots + X_n$.

Let $X_1, X_2, \ldots, X_n$ be i.i.d. uniform random variables on [0, 1].

Let $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$ be the order statistics.

(1) Find the density of $X_{(k)}$, the k-th order statistic.

(2) Find $E[X_{(k)}]$.

(3) Find the joint density of $(X_{(1)}, X_{(n)})$.

Let $X_n$ have distribution function:

(1) Find the limiting distribution as n → ∞.

(2) Identify the limit distribution.

Let N ~ Poisson(λ), and given N = n, let $X_1, \ldots, X_N$ be i.i.d. with mean μ and variance σ².

Define $S = \sum_{i=1}^N X_i$ (with S = 0 if N = 0).

(1) Find E[S].

(2) Find Var(S).

Let X ~ Uniform(0, 1). Find the distribution of:

(1) $Y = -\ln(X)$

(2) $Z = X^2$

(3) $W = \frac{X}{1-X}$