Recent questions tagged probability-and-statistics

Consider a discrete time channel with binary inputs and binary outputs. Let $x_{n}$ denote the input bit at time $n$ and $y_{k}$ denote the output bit at time $\text{k}$....

Define $\operatorname{sign}(x)=0$ for $x=0, \operatorname{sign}(x)=1$ for $x>0$ and $\operatorname{sign}(x)=-1$ for $x<0$. For $n \geq 0$, let\[Y_{n}=\operatorname{sign}\...

Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. The probability that $\text{X + ...

Suppose that a random variable $X$ can take $5$ values $\{1,2,3,4,5\}$ with probabilities that depend upon $n \geq 0$ and are given by\[P(X=k)=\frac{e^{k n}}{e^{n}+e^{2 n...

Consider a coin flip game between Amar, Akbar and Anthony. A fair coin (so that heads and tails each have probability $0.5)$ is independently flipped five times. Amar win...

Two players $\mathrm{A}$ and $\mathrm{B}$ of equal skill are playing a match. The first one to win $4$ rounds wins the match. Both players are equally likely to win each ...

Suppose you throw a dart and it lands uniformly at random on a target which is a disk of unit radius. What is the probability density function $f(x)$ of the distance of t...

A drunken man walks on a straight lane. At every integer time (in seconds) he moves a distance of $1$ unit randomly, either forwards or backwards. What is the expectation...

Let a bag contain ten balls numbered $1,2, \ldots, 10$. Let three balls be drawn at random in sequence without replacement, and the number on the ball drawn on the $i^{\t...

Consider a fair coin (i.e., both heads and tails have equal probability of appearing). Suppose we toss the coin repeatedly until both sides have been seen. What is the ex...

A stick of length $1$ is broken at a point chosen uniformly at random. Which of the following is false?Twice the length of the smaller piece is greater than the length of...

Suppose that $X_{1}$ and $X_{2}$ denote the output of rolls of two independent dices that can each take integer values $\{1,2,3,4,5,6\}$ with probability $1 / 6$ for each...

A tourist starts by taking one of the $n$ available paths, denoted by $1,2, \cdots, n$. An hour into the journey, the path $i$ subdivides into further $1+i$ subpaths, onl...

We have the sequence, $a_{n}=\frac{1}{n \log ^{2} n}, n \geq 2$, where log is the logarithm to the base $2$ and let $A=\sum_{n=2}^{\infty} a_{n}$ be the sum of the sequen...

Balls are drawn one after the other uniformly at random without replacement from a set of eight balls numbered $1,2, \ldots, 8$ until all balls drawn. What is the expecte...

Given $n$ independent Bernoulli random variables, taking value $1$ with probability $p$ and $0$ with probability $1-p$. Then, which of the following is the value of $E\le...

Suppose that Dice $1$ has five faces numbered $1$ to $5,$ each of which is equally likely to occur once the dice is rolled. Dice $2$ similarly has eight equally likely fa...

Consider two independent random variables $\left(U_{1}, U_{2}\right)$ both are uniformly distributed between $[0,1]$. The conditional expectation\[E\left[\left(U_{1}+U_{2...

Suppose that $X$ is a real valued random variable and $E[\exp X]=2$. Then, which of the following must be $\text{TRUE? Hint:}$ $(\exp (x)+\exp (y)) / 2 \geq \exp ((x+y) /...

Consider a unit disc $D$. Let a point $x$ be chosen uniformly on $D$ and let the random distance to $x$ from the center of $D$ be $R$. Which of the following is $\text{TR...

Alice and Bob have one coin each with probability of Heads $p$ and $q$, respectively. In each round, both Alice and Bob independently toss their coin once, and the game s...

Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice. Each of the dice takes each of the six values $1,2,3,4,5$, and $6$ with equa...

Consider two random variables $X$ and $Y$ which take values in a finite set $S$. Let $p_{X, Y}$ represent their joint probability mass function (p.m.f.) and let $p_{X}$ a...

Consider a coin which comes up heads with probability $p$ and tails with probability $1-p$, where $0 < p < 1.$ Suppose we keep tossing the coin until we have seen both si...

Let $X, Z_{1}$, and $Z_{2}$ be independent random variables taking values in the set $\{0,1\}$. $X$ is uniformly distributed in $\{0,1\}$, while the distributions of $Z_{...

Let $X$ and $Y$ be independent Gaussian random variables with means $1$ and $2$ and variances $3$ and $4$ respectively. What is the minimum possible value of $\mathbf{E}\...

Consider an urn with $a$ red and $b$ blue balls. Balls are drawn out one-by-one, without replacement and uniformly at random, until the first red ball is drawn. What is t...

For $t>0$, let $S_{t}$ denote the ball of radius $t$ centered at the origin in $\mathbb{R}^{n}$. That is,\[S_{t}=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \sum_{i=1}^{n} ...

Consider the circle of radius $1$ centred at the origin in two dimensions. Choose two points $x$ and $y$ independently at random so that both are uniformly distributed on...

Anu reached a bus stop at $\text{9:00 AM.}$ She knows that the number of minutes after $\text{9:00 AM}$ when the bus will arrive is distributed with probability density f...