Recent questions tagged random-variable

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 $Y$ be a continuous random variable with density,$$f_Y(y)= \begin{cases}2y, & \text { if } 0<y<1 \\\\ 0, & \text { otherwise }\end{cases}$$Given $Y=y$, the conditiona...

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...

Let $X$ and $Y$ be independent standard normal random variables. Define $A=X+Y$ and $B=X-Y$. Find $P(|A|<B)$.$1/4$ $1/2$ $3/4$ $1/3$

Let $X$ be an exponentially distributed random variable with mean $1/\ln 3$. Define $Y=\lfloor X \rfloor$, where $\lfloor X \rfloor$ is the greatest integer less than or ...

A biased coin is flipped $6$ times, and the probability of getting a head on a single flip is $1/3$. Given that the total number of heads is even, what is the probability...

Suppose $X,Y,Z,W$ are independent random variables taking values $0$ or $1$ with equal probability $1/2$. Find $P(X+Y+Z+W=2 \mid X=Y \text{ or } Z=W)$.$1/2$ $1/3$ $1/6$ $...

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 website assigns each new visitor one of $6$ equally likely security icons, independently of all previous visitors. The administrator watches the sequence of assigned ic...

A random variable $X$ has probability mass function given by :$$p_X(x)=\begin{cases} c, & \text{if } x=-1 \\\\ 2c, & \text{if } x=1 \\\\ \frac{1}{6}, & \text{if } x=4 \\\...

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...

Suppose you keep tossing a fair coin until you get the first head. Let $Y$ be the number of tosses until the first head appears. Find the probability that $Y$ is odd, giv...

Suppose that $X$ and $Y$ are independent random variables, such that :$P(X=x)=\frac{1}{4}\left(\frac{3}{4}\right)^{x-1}$ for integers $x\ge 1$, and $P(Y=y)=\frac{1}{2}\le...

Let $X_1,X_2,\ldots,X_n$ be independent discrete uniform random variables on ${1,2,\ldots,n}$. That is, $P(X_i=j)=\frac{1}{n}$ for each $i$ and each $j\in{1,2,\ldots,n}$....

Suppose that $X$, $Y$, and $Z$ are independent random variables such that : $P(X=x)=\frac{1}{2}\left(\frac{1}{2}\right)^{x-1}$ for integers $x\ge 1$,  $P(Y=y)=\frac{1}{3}...

Let $X_1,X_2,X_3$ be independent continuous uniform random variables on $[0,1]$. Let $k\in[0,1]$ be a constant. Find $P(\max(X_1,X_2,X_3)>k)$.$1-k$ $1-k^2$ $1-k^3$ $(1-k)...

Let $X$ be a random variable with PDF :$$f(x)= \begin{cases}C x^2(1-x)^3 & 0 <x<1\\\\ 0 & \text{otherwise} \end{cases}$$What is $C$?

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 $X$, $Y$, and $Z$ be random variables which are not necessarily independent. Suppose $Var(X)=4$, $Var(Y)=9$, $Var(Z)=16$, $Var(X+Y+Z)=39$, $Var(X-Y+Z)=11$, and $Var(2...

Let $X$ and $Y$ be discrete random variables. $X$ takes values $0,1,2,3$ and $Y$ takes values $1,2,3,4$. Their joint probability mass function is given below.$$\begin{arr...

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]...

Let $X$ and $Y$ be non-negative random variables, which may or may not be independent. Which of the following statements is/are ALWAYS true?$P(X+Y\ge 2)\ge P(X\ge 1)P(Y\g...

A number $X$ is chosen from ${1,2,3,4,5,6}$ with $P(X=x)=x/21$.After $X=x$ is chosen, a second number $Y$ is chosen uniformly from ${1,2,\ldots,x}$.Given this setup, whic...

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...

A reliability engineer tests two independent machines. Machine $A$ has $7$ identical safety sensors, and machine $B$ has $11$ identical safety sensors. Each sensor indepe...

In a cyber-security control room, six independent scanners report integer threat scores. Scanner $A$ reports uniformly from $1$ to $5$, scanner $B$ reports uniformly from...

Let $X$ and $Y$ be jointly continuous random variables with joint density function : $$f\left(x, y\right)= \begin{cases}cx y(1+y) & \text { if } 0 \leq x \leq 1 \text { a...

Consider three Bernoulli random variables $X$, $Y$, and $Z$, each taking values $0$ or $1$. No independence assumptions are made. Suppose$P(X=1)=\frac{2}{5}$,  $P(Y=1|X=1...