Comprehensive Foundations Of Probability And Stochastic Processes Practice Tests

Question 1

Consider two events $A$ and $B$ in a sample space $\Omega$ such that $P(A) = 0.6$, $P(B) = 0.4$, and $P(A \cup B) = 0.8$. Calculate the conditional probability $P(A | B)$.

Accepted Answer

0.5

Question 2

Let $X$ be a continuous random variable with the cumulative distribution function $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$ and $F(x) = 0$ for $x < 0$. What is the probability density function $f(x)$ for $x \geq 0$?

Accepted Answer

Option A

Answer

Option B

Answer

Option C

Answer

Option D

Question 3

If two events $A$ and $B$ are disjoint, they must be independent.

Accepted Answer

False

Answer

True

Question 4

Which of the following conditions are required for a function $F(x) = P(X \leq x)$ to be a valid cumulative distribution function?

Accepted Answer

Option A

Answer

Option B

Answer

Option C

Answer

Option D

Question 5

A discrete random variable $X$ has a probability mass function $f(x) = c \cdot x$ for $x \in \{1, 2, 3, 4\}$. Determine the value of the constant $c$.

Accepted Answer

0.1

Question 6

According to the law of large numbers, as the number of independent trials $n$ increases, what does the sample average $\bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i$ converge to?

Accepted Answer

Option A

Answer

Option B

Answer

Option C

Answer

Option D

Question 7

For a discrete random variable $X$ with probability mass function $P(X=k) = \frac{1}{3}$ for $k \in \{1, 2, 3\}$, calculate the variance $Var(X) = E(X^2) - (E(X))^2$.

Accepted Answer

2/3

Question 8

Let $X$ and $Y$ be independent discrete random variables. Which of the following conditions must hold for their joint probability mass function $p(x, y)$?

Accepted Answer

$p(x, y) = P(X=x)P(Y=y)$

Answer

$p(x, y) = P(X=x) + P(Y=y)$

Answer

$p(x, y) = \frac{P(X=x)}{P(Y=y)}$

Answer

$p(x, y) = E(X)E(Y)$

Question 9

In a simple symmetric random walk starting at $0$, what is the probability that the walk is at position $0$ after $2n$ steps?

Accepted Answer

$\binom{2n}{n} (1/2)^{2n}$

Question 10

Given a continuous random variable $X$ with density $f(x) = 2x$ for $0 \leq x \leq 1$, find the expected value $E[X]$.

Accepted Answer

$2/3$

Answer

$1/2$

Answer

$3/4$

Answer

$1$

Question 11

If $Y = X^2$ where $X$ is a continuous random variable with density $f_X(x)$, express the density $f_Y(y)$ for $y > 0$ using the transformation method.

Accepted Answer

$\frac{1}{2\sqrt{y}} (f_X(\sqrt{y}) + f_X(-\sqrt{y}))$

Question 12

For a random variable $X$ with probability generating function $G(s) = E(s^X)$, calculate the mean $E[X]$ in terms of $G(s)$.

Accepted Answer

$G'(1)$

Question 13

In a Galton-Watson branching process with offspring distribution mean $\mu < 1$, what is the probability of ultimate extinction?

Accepted Answer

$1$

Answer

$0$

Answer

$0.5$

Answer

$\mu$

Question 14

Consider a Markov chain with state space $S = \{1, 2, 3\}$ and transition matrix $P = \begin{pmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0 \end{pmatrix}$. Determine the stationary distribution $\pi = (\pi_1, \pi_2, \pi_3)$.

Accepted Answer

(\frac{1}{3}, \frac{1}{3}, \frac{1}{3})

Answer

(1, 0, 0)

Answer

(\frac{1}{2}, \frac{1}{4}, \frac{1}{4})

Answer

(0, 0, 1)

Question 15

Let $X_n$ be a sequence of i.i.d. random variables with mean $\mu = 2$ and variance $\sigma^2 = 4$. Using the Central Limit Theorem, estimate the probability $P(\sum_{i=1}^{100} X_i > 220)$.

Accepted Answer

0.1587

Question 16

If a sequence of random variables $X_n$ converges to $X$ in the $r$-th mean (for $r \geq 1$), then $X_n$ also converges to $X$ in probability.

Accepted Answer

True

Answer

False

Question 17

Which of the following conditions are necessary for a Markov chain to possess a unique stationary distribution?

Answer

The chain is irreducible

Answer

The chain is aperiodic

Answer

The chain is positive recurrent

Answer

The chain has a finite state space

Question 18

A Wiener process $W_t$ satisfies $W_0 = 0$. Calculate the covariance $Cov(W_s, W_t)$ for $0 \leq s \leq t$.

Accepted Answer

s

Question 19

For a stationary process $X_t$ with autocovariance function $R(	au) = E[(X_{t+	au} - \mu)(X_t - \mu)]$, what is the relationship between $R(	au)$ and $R(-	au)$?

Accepted Answer

R(	au) = R(-	au)

Answer

R(	au) = -R(-	au)

Answer

R(	au) = |R(-	au)|

Answer

R(	au) = R(-	au)^2

Question 20

In a renewal process with interarrival times $X_i$ having mean $E[X_i] = \mu = 5$, determine the limit of the renewal function $m(t) = E[N(t)]$ as $t 	o \infty$ relative to $t$.

Accepted Answer

1/5

Comprehensive Foundations Of Probability And Stochastic Processes Practice Tests

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Comprehensive Foundations of Probability and Stochastic Processes

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