# Tagged: expected value

## Problem 759

(a) Let $X$ be a random variable that takes only non-negative values. Prove that for any $a > 0$,
$P(X \geq a) \leq \frac{E[X]}{a}.$ This inequality is called Markov’s inequality.

(b) Let $X$ be a random variable with finite mean $\mu$ and variance $\sigma^2$. Prove that for any $a >0$,
$P\left(|X – \mu| \geq a \right) \leq \frac{\sigma^2}{a^2}.$ This inequality is called Chebyshev’s inequality.

## Problem 757

Let $X$ be an exponential random variable with parameter $\lambda$.

(a) For any positive integer $n$, prove that
$E[X^n] = \frac{n}{\lambda} E[X^{n-1}].$

(b) Find the expected value of $X$.

(c) Find the variance of $X$.

(d) Find the standard deviation of $X$.

## Problem 753

Let $c$ be a fixed positive number. Let $X$ be a random variable that takes values only between $0$ and $c$. This implies the probability $P(0 \leq X \leq c) = 1$. Then prove the next inequality about the variance $V(X)$.
$V(X) \leq \frac{c^2}{4}.$

## Problem 750

A box of some snacks includes one of five toys. The chances of getting any of the toys are equally likely and independent of the previous results.

(a) Suppose that you buy the box until you complete all the five toys. Find the expected number of boxes that you need to buy.

(b) Find the variance and the standard deviation of the event in part (a).

## Problem 749

A final exam of the course Probability 101 consists of 10 multiple-choice questions. Each question has 4 possible answers and only one of them is a correct answer. To pass the course, 8 or more correct answers are necessary. Assume that a student has not studied probability at all and has no idea how to solve the questions. So the student decided to answer each questions randomly. Thus, for each of 10 questions, the student choose one of the 4 answers randomly and each choice is independent each other.

(1) What is the probability that the student answered correctly only one question among the 10 questions?

(2) Determine the probability that the student passes the course.

(3) What is the expected value of the number of questions the student answered correctly?

(4) Find the variance and standard deviation of the number of questions the student answered correctly.

## Problem 748

Suppose that $X$ is a random variable with Bernoulli distribution $B_p$ with probability parameter $p$.

Assume that the variance $V(X) = 0.21$. We further assume that $p > 0.5$.

(a) Find the probability $p$.

(b) Find the expectation $E(X)$.

## Problem 747

A random variable $X$ is said to be a Bernoulli random variable if its probability mass function is given by
\begin{align*}
P(X=0) &= 1-p\\
P(X=1) & = p
\end{align*}
for some real number $0 \leq p \leq 1$.

(1) Find the expectation of the Bernoulli random variable $X$ with probability $p$.

(2) Find the variance of $X$.

(3) Find the standard deviation of $X$.

## Problem 743

Let $X, Y$ be discrete random variables. Prove the linearity of expectations described as
$E(X+Y) = E(X) + E(Y).$