Recent questions tagged expectation

A machine produces a random score $T$ between $0$ and $2$ with probability density function $$f(t)= \begin{cases}\frac{3t^2}{8}, & \text { if } 0<t<2 \\\\ 0, & \text { ot...

Let $X$ be a continuous random variable on $[0,1]$ with probability density function $f(x)=3x^2$ for $0\le x\le 1$ and $0$ otherwise. Let $Y=\log_e(1+X^3)$. Find the expe...

A game is played with a weighted coin that lands heads with probability $3/5$ and tails with probability $2/5$. The game has normal rounds and bonus rounds. The first rou...

Let $X_1,X_2,X_3$ be independent Bernoulli random variables with parameter $p$. Find $E[X_1^2X_2(2X_1+X_3)]$.$2p^2+p^3$ $3p^3$ $2p^3+p^2$ $p^2+p^3$

A delivery worker is assigned one of three types of delivery routes by drawing a card from a box containing $6$ cards numbered $1$ through $6$.If the card is $1$ or $2$, ...

Suppose that a person plays $12$ independent games. In each game, he wins with probability $\frac{1}{3}$ and loses otherwise. If he wins a game, he earns $9$ dollars, and...

Let $X$ be a Bernoulli random variable with parameter $p$, where $0<p<1$. Define $Y=5^{X}2^{1-X}+3X$. What is $E[Y]$?$2+6p$ $5p+2(1-p)$ $2+3p$ $5+2p$

Let $X$ and $Y$ be independent discrete uniform random variables on ${1,2,\dots,n}$. Let $H_n=1+\frac12+\frac13+\cdots+\frac1n$. The value of $E\left[\frac{1}{X+Y}\right]...

Suppose $X$, $Y$, and $Z$ are independent random variables with $E[X]=2$, $E[Y]=-1$, $E[Z]=3$, $\operatorname{Var}(X)=5$, $\operatorname{Var}(Y)=4$, and $\operatorname{Va...

There are $12$ cards in a hat. For each rank $1,2,3,4$, there are exactly $3$ cards of that rank. You draw one card at random. If the first card has rank $1$, you draw $1...

Let $X$ be a random variable such that $P(X=-2)=\frac{1}{4}$, $P(X=0)=\frac{1}{2}$, and $P(X=2)=\frac{1}{4}$.What is the value of $E[X^2]-{E[X]}^2$ ?

Suppose that $X$ is a random variable where $P(X=-4)=c$, $P(X=-1)=2c$, $P(X=2)=3c$, and $P(X=5)=4c$, where $c>0$. Let $Y=(X-E[X])^2$. What is the value of $E[Y]$?$7$ $8$ ...

James Bond repeatedly rolls a fair standard $6$-sided die. What is the expected number of rolls until he gets two consecutive $5$'s for the first time?$36$ $40$ $42$ $48$

Let $X$ be a random variable which takes values in the set $\{1,2,3,4,5,6,7,8\}$. Further, $\operatorname{Pr}(X=1)=\operatorname{Pr}(X=2)=\operatorname{Pr}(X=5)=\operator...

$$\begin{aligned}P(X=1)=P(X=2)=P(X=5)=P(X=7) &= \frac{1}{6} \\P(X=3)=P(X=4)=P(X=6)=P(X=8) &= \frac{1}{12}\end{aligned}$$What is the expected value $E[X]$?

$15$ balls are placed independently and uniformly at random into $15$ bins numbered from $1$ to $15.$ The probability that a ball ends up in a particular bin is $\frac{1}...

Let $X$ and $Y$ be continuous random variables with joint PDF: $f_{X, Y}(x, y)= \begin{cases}k & \text { if } 0<y \leq x \leq 1 \\ 0 & \text...

A game being offered in a casino consists of guessing the outcomes of two tosses of a fair coin. The gambler wins if she/he has correctly guessed at least one of the two ...

$15$ balls are placed independently and uniformly at random into $15$ bins numbered from $1$ to $15.$ The probability that a ball ends up in a particular bin is $\frac{1}...

Consider a biased coin which has probability $p$ of turning up heads (and $1-p$ of turning up tails), and consider an experiment where we toss the coin repeatedly. What i...

Consider the collection $\mathcal{C}$ of all possible $n \times n$ matrices with entries in $\{1, \ldots, n\}$ that obey the following property: every row of $M$ is a per...

Consider a coin which has probability $p$ of coming up heads when tossed. Assume that $0<p<10 < p < 10<p<1$. A trial is performed by repeatedly tossing the coin until the...

Suppose $3$ elements are hashed independently and uniformly at random one by one to slots in a hash table of size $6$ (assume that in case of a collision, the element is ...