Recent questions tagged probability-and-statistics

The following observation is made about the scores obtained by $100$ students in an exam:'For each student, there exists another student in the class such that their scor...

Let $X, N, Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between $-1$ and $1 ; N$ follows Normal d...

A QPSK modulated signal from an additive white Gaussian noise (AWGN) channel is received with an $E_{b} / N_{o}=8.4 \mathrm{~dB}$ at the input of a coherent QPSK demodula...

Let the relevant bandwidth $(B)$ of a digital communication system be $1 ~\text{MHz}$ and $k T=-174 \mathrm{~dBm} / \mathrm{Hz}$, where $k$ is Boltzmann's constant and '$...

The average bit error rate at the input of a $(7,4,1)$ Hamming decoder is $0.10$. The probability that the decoder will fail to decode a received word correctly is $\_\_\...

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement.What is the probability that the two balls drawn have diff...

A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where $N$ is a zero-mean Gaussian random vari...

Consider a real-valued random process$$f(t)=\sum_{n=1}^{N} a_{n} \: p(t-n T),$$where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and $0$ othe...

The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)=\alpha$ and $P(X=0)=1-2 \alpha$, where $0<\alpha<1 / 2$.Let $g(\alpha)$ denote the...

Two fair dice (with faces labeled $1,2,3,4,5$, and $6$) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained.The expectation of $X$ is ________...

Suppose $\text{X}$ and $\text{Y}$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability tha...

​​​​​A source transmits a symbol $s$, taken from $\{-4,0,4\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy symbol $r$ is give...

For a real signal, which of the following is/are valid power spectral density/densities?$\text{S}_X(\omega)=\frac{2}{9+\omega^2}$$\text{S}_X(\omega)=e^{-\omega^2} \cos ^2...

A random variable $\mathrm{X}$, distributed normally as $\mathrm{N(0,1)}$ undergoes the transformation$\mathrm{Y}=\mathrm{h}(\mathrm{X})$, given in the figure. The form o...

The frequency of occurrence of $8$ symbols $(a-h)$ is shown in the table below. A symbol is chosen and it is determined by asking a series of $\text{"yes/no"}$ questions ...

Consider a fair coin with probability of heads and tails equal to $1 / 2$. Moreover consider two dice, first $\mathrm{D}_{1}$ that has three faces numbered $1,3,5$ and se...

Recall that the entropy (in bits) of a random variable $\mathrm{X}$ which takes values in $\mathbb{N}$, the set of natural numbers, is defined as$$H(X)=\sum_{n=1}^{\infty...

Let $f(x)$ be a positive continuous function on the real line that is the density of a random variable $X$. The differential entropy of $X$ is defined to be $-\int_{-\inf...

Suppose a bag contains $5$ red balls, $3$ blue balls, and $2$ black balls. Balls are drawn without replacement until the bag is empty. Let $X_{i}$ be a random variable wh...

Consider a disk $D$ of radius $1$ centered at the origin. Let $X$ be a point uniformly distributed on $D$ and let the distance of $X$ from the origin be $R$. Let $A$ be t...

Let $X$ be a random variable which takes values $1$ and $-1$ with probability $1 / 2$ each. Suppose $Y=X+N$, where $N$ is a random variable independent of $X$ with the fo...

The capacity of a certain additive white Gaussian noise channel of bandwidth $1 \mathrm{~MHz}$ is $\mathrm{known}$ to be $8 \text{ Mbps}$ when the average transmit power ...

Let $X$ and $Y$ be two independent and identically distributed random variables. Let $Z=\max (X, Y)$ and $W=\min (X, Y)$. Which of the following is true?$Z$ and $W$ are i...

Consider a random variable $X$ that takes integer values $1$ through $10$ each with equal probability. Now consider random variable\[Y=\min (7, \max (X, 4)).\]What is the...

Let $X$ be a uniform random variable between $[0,1]$. And let\[M=\min _{m X \geq 1, m \in \mathbb{N}} m .\]Then which of the following is true?$E(M)=\infty$$E(M) \in[5,10...

Consider a frog that lives on two rocks $A$ and $B$ and moves from one rock to the other randomly. If it is at Rock $A$ at any time, irrespective of which rocks it occupi...

Let $x_{1}=-1$ and $x_{2}=1$ be two signals that are transmitted with equal probability. If signal $x_{i}, i \in$ $\{1,2\}$ is transmitted, the received signal is $y=x_{i...