Most viewed questions in Probability and Statistics / GO Electronics

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81

TIFR ECE 2015 | Question: 15
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 $y=x_{i}+n_{i}$, where $n_{i}$ ... $\theta^{\star}$ to minimize the probability of error is $\leq 0$ None of the above.

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...

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82

TIFR ECE 2012 | Question: 14
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is \[f_{X}(x)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{2}\right),-\infty<x<\infty. \] The random variable $Z$ is defined ... none of the above

Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is\[f_{X}(x)=\frac{1}{\sqrt{2 \pi}}...

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83

TIFR ECE 2012 | Question: 10
Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the average of the other two outcomes. Here, the ties may be resolved arbitrarily. $1 / 6$ $1 / 9$ $39 / 216$ $7 / 36$ $43 / 216$

Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the ...

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84

TIFR ECE 2022 | Question: 1
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 \[P(X=k)=\frac{e^{k n}}{e^{n}+e^{2 n}+e^{3 n}+e^{4 n}+e^{5 n}}\] for $k=1,2,3,4,5$. ... $1$ as $n \rightarrow \infty$ It converges to $5$ as $n \rightarrow \infty$ It converges to $0$ as $n \rightarrow \infty$

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...

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85

GATE ECE 2014 Set 4 | Question: 50
Consider the $Z$-channel given in the figure. The input is $0$ or $1$ with equal probability. If the output is $0$, the probability that the input is also $0$ equals ___________

Consider the $Z$-channel given in the figure. The input is $0$ or $1$ with equal probability.If the output is $0$, the probability that the input is also $0$ equals _____...

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86

GATE ECE 2014 Set 3 | Question: 28
A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is _______ .

A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is _______ .

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87

TIFR ECE 2013 | Question: 4
Consider a fair coin that has probability $1 / 2$ of showing heads $(\text{H})$ in a toss and $1 / 2$ of showing tails $(\text{T})$. Suppose we independently flip a fair coin over and over again. What is the probability that $\text{HT}$ sequence occurs before $\text{TT}?$ $3 / 4$ $1 / 2$ $2 / 3$ $1 / 3$ $1 / 4$

Consider a fair coin that has probability $1 / 2$ of showing heads $(\text{H})$ in a toss and $1 / 2$ of showing tails $(\text{T})$. Suppose we independently flip a fair ...

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88

TIFR ECE 2022 | Question: 2
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 wins if at least three consecutive draws of heads are observed in the five coin tosses. Akbar wins if at least three ... What is the probability of Anthony winning? $9 / 16$ $1 / 3$ $1 / 2$ $5 / 8$ $7 / 12$

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...

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89

TIFR ECE 2020 | Question: 10
Consider two independent random variables $\left(U_{1}, U_{2}\right)$ both are uniformly distributed between $[0,1]$. The conditional expectation \[E\left[\left(U_{1}+U_{2}\right) \mid \max \left(U_{1}, U_{2}\right) \geq 0.5\right]\] equals $7 / 6$ $8 / 7$ $6 / 7$ $1.1$ None of the above

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...

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90

TIFR ECE 2017 | Question: 12
Consider a signal $X$ that can take two values, $-1$ with probability $p$ and $+1$ with probability $1-p$. Let $Y=X+N$, where $N$ is mean zero random noise that has probability density function symmetric about $0.$ Given $p$ and on observing $Y$, the detection problem is ... $\text{(iii)}$ Only $\text{(i)}$ and $\text{(ii)}$ Only $\text{(i)}$ and $\text{(iii)}$

Consider a signal $X$ that can take two values, $-1$ with probability $p$ and $+1$ with probability $1-p$. Let $Y=X+N$, where $N$ is mean zero random noise that has proba...

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91

TIFR ECE 2016 | Question: 2
Let $X_{1}$ and $X_{2}$ be two independent continuous real-valued random variables taking values in the unit interval $[0,1]$. Let $Y=\max \left\{X_{1}, X_{2}\right\}$ ... $\operatorname{Pr}[Z=1]>\operatorname{Pr}[Z=2]=\frac{1}{2}$ $\operatorname{Pr}[Z=1]<\operatorname{Pr}[Z=2]$

Let $X_{1}$ and $X_{2}$ be two independent continuous real-valued random variables taking values in the unit interval $[0,1]$. Let $Y=\max \left\{X_{1}, X_{2}\right\}$ an...

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92

TIFR ECE 2013 | Question: 9
Let $X$ and $Y$ be two zero mean independent continuous random variables. Let $Z_{1}=\max (X, Y)$, and $Z_{2}=\min (X, Y)$. Then which of the following is TRUE. $Z_{1}$ and $Z_{2}$ are uncorrelated. $Z_{1}$ and $Z_{2}$ are independent. $P\left(Z_{1}=Z_{2}\right)=\frac{1}{2}$. Both $(a)$ and $(c)$ Both $(a)$ and $(b)$

Let $X$ and $Y$ be two zero mean independent continuous random variables. Let $Z_{1}=\max (X, Y)$, and $Z_{2}=\min (X, Y)$. Then which of the following is TRUE.$Z_{1}$ an...

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93

TIFR ECE 2016 | Question: 10
Let $U_{1}, U_{2}, U_{3}$ be independent random variables that are each uniformly distributed between zero and one. What is the probability that the second highest value amongst the three lies between $1 / 3$ and $2 / 3?$ $\frac{2}{9}$ $\frac{1}{27}$ $\frac{13}{27}$ $\frac{1}{3}$ $\frac{7}{18}$

Let $U_{1}, U_{2}, U_{3}$ be independent random variables that are each uniformly distributed between zero and one. What is the probability that the second highest value ...

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94

GATE ECE 2014 Set 1 | Question: 49
Let $X$ be a real-valued random variable with $E[X]$ and $E[X^{2}]$ denoting the mean values of $X$ and $X^{2},$ respectively. The relation which always holds true is $(E[X])^{2}>E[X^{2}]$ $E[X^{2}]\geq (E[X])^{2}$ $E[X^{2}] = (E[X])^{2}$ $E[X^{2}] > (E[X])^{2}$

Let $X$ be a real-valued random variable with $E[X]$ and $E[X^{2}]$ denoting the mean values of $X$ and $X^{2},$ respectively. The relation which always holds true is$(E[...

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95

TIFR ECE 2017 | Question: 11
Consider a unit length interval $[0,1]$ and choose a point $X$ on it with uniform distribution. Let $L_{1}=X$ and $L_{2}=1-X$ be the length of the two sub-intervals created by this point on the unit interval. Let $L=\max \left\{L_{1}, L_{2}\right\}$. Consider ... $\text{(ii)}$ Only $\text{(i)}$ and $\text{(iii)}$ Only $\text{(ii)}$ and $\text{(iv)}$ None of the above

Consider a unit length interval $[0,1]$ and choose a point $X$ on it with uniform distribution. Let $L_{1}=X$ and $L_{2}=1-X$ be the length of the two sub-intervals creat...

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96

TIFR ECE 2016 | Question: 9
Suppose $Y=X+Z$, where $X$ and $Z$ are independent zero-mean random variables each with variance $1.$ Let $\hat{X}(Y)=a Y$ be the optimal linear least-squares estimate of $X$ from $Y$, i.e., $a$ is chosen such that $E\left[(X-a Y)^{2}\right]$ is minimized. What is the resulting ... $1$ $\frac{2}{3}$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$

Suppose $Y=X+Z$, where $X$ and $Z$ are independent zero-mean random variables each with variance $1.$ Let $\hat{X}(Y)=a Y$ be the optimal linear least-squares estimate of...

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97

TIFR ECE 2021 | Question: 15
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 $A=\sum_{n=2}^{\infty} a_{n}$ ... $H(X)?$ $H(X) \leq 3$ $H(X) \in(3,5]$ $H(X) \in(5,10]$ $H(X)>10$ but finite $H(X)$ is unbounded

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...

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98

TIFR ECE 2012 | Question: 11
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales as $\lambda$ $\lambda^{2}$ $\lambda^{3}$ $\sqrt{\lambda}$ None of the above

A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales ...

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99

TIFR ECE 2021 | Question: 14
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, only one of which leads to the destination. The tourist has no map and makes random choices of the path and the ... $\frac{10}{36}$ $\frac{11}{36}$ $\frac{12}{36}$ $\frac{13}{36}$ $\frac{14}{36}$

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...

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100

TIFR ECE 2017 | Question: 9
Recall that for a random variable $X$ which takes values in $\mathbb{N}$, the set of natural numbers, its entropy in bits is defined as \[H(X)=\sum_{n=1}^{\infty} p_{n} \log _{2} \frac{1}{p_{n}},\] where, for $n \in \mathbb{N}, p_{n}$ denotes the ... entropy of $X$ in bits? $1$ $1.5$ $\frac{1+\sqrt{5}}{2} \approx 1.618$ (the golden ratio) $2$ None of the above

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

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101

TIFR ECE 2013 | Question: 16
A surprise quiz contains three multiple choice questions; question $1$ has $3$ suggested answers, question $2$ has four, and question $3$ has two. A completely unprepared student decides to choose the answers at random. If $X$ is the number of questions the student answers ... expected number of correct answers is $15 / 12$ $7 / 12$ $13 / 12$ $18 / 12$ None of the above

A surprise quiz contains three multiple choice questions; question $1$ has $3$ suggested answers, question $2$ has four, and question $3$ has two. A completely unprepared...

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admin asked Dec 12, 2022

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102

GATE ECE 2010 | Question: 27
A fair coin is tossed independently four times. The probability of the event "the number of times heads show up is more than the number of times tails show up" is $\frac{1}{16}$ $\frac{1}{8}$ $\frac{1}{4}$ $\frac{5}{16}$

A fair coin is tossed independently four times. The probability of the event "the number of times heads show up is more than the number of times tails show up" is$\frac{1...

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admin asked Sep 15, 2022

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103

TIFR ECE 2020 | Question: 13
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 stops if one of them gets a Heads and the other gets a Tails. If they both get either Heads or both get Tails in ... $R$ is independent of $p$ and $q$ $R=\frac{1}{1+2 p q-p-q}$ None of the above

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...

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104

GATE ECE 2011 | Question: 36
A fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is $2 / 36$ $2 / 6$ $5 / 12$ $1 / 2$

A fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is$2 / 36$$2 / 6$$5 / 12$$1 / 2$

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105

TIFR ECE 2013 | Question: 18
Consider a coin tossing game between Santa and Banta. Both of them toss two coins sequentially, first Santa tosses a coin then Banta and so on. Santa tosses a fair coin: Probability of heads is $1 / 2$ and probability of tails is $1 / 2$. Banta's coin probabilities depend on ... the two trials conducted by each of them? $1 / 2$ $5 / 16$ $3 / 16$ $1 / 4$ $1 / 3$

Consider a coin tossing game between Santa and Banta. Both of them toss two coins sequentially, first Santa tosses a coin then Banta and so on. Santa tosses a fair coin: ...

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admin asked Dec 12, 2022

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106

TIFR ECE 2021 | Question: 9
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 the larger piece with positive probability. One half of the length of the ... . The product of the length of the smaller piece and the larger piece is greater than $1 / 4$ with positive probability.

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...

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107

TIFR ECE 2013 | Question: 14
$X, Y, Z$ are integer valued random variables with the following two properties: $X$ and $Y$ are independent. For all integer $x$, conditioned on the event $\{X=x\}$, we have that $Y$ and $Z$ are independent (in other words, conditioned on ... and $Z$ are independent Conditioned on $Z$, the random variables $X$ and $Y$ are independent All of the above None of the above

$X, Y, Z$ are integer valued random variables with the following two properties:$X$ and $Y$ are independent.For all integer $x$, conditioned on the event $\{X=x\}$, we ha...

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108

TIFR ECE 2013 | Question: 17
Consider four coins, three of which are fair, that is they have heads on one side and tails on the other and both are equally likely to occur in a toss. The fourth coin has heads on both sides. Given that one coin amongst the four is picked at random and is tossed, and the ... is the probability that its other side is tails? $1 / 2$ $3 / 8$ $3 / 5$ $3 / 4$ $5 / 7$

Consider four coins, three of which are fair, that is they have heads on one side and tails on the other and both are equally likely to occur in a toss. The fourth coin h...

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admin asked Dec 12, 2022

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109

TIFR ECE 2020 | Question: 12
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 $\text{TRUE?}$ $R^{2}$ is uniformly distributed in $[0,1]$ $\pi R^{2}$ is uniformly ... $[0,1]$ $2 \pi R^{2}$ is uniformly distributed in $[0,1]$ None of the above

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...

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110

TIFR ECE 2014 | Question: 16
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is $4 \frac{1}{2}$ $3 \frac{1}{2}$ $5$ $4 \frac{17}{36} $ $4 \frac{3}{4}$

A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is$4 \frac{1}{2...

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admin asked Dec 14, 2022

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111

TIFR ECE 2019 | Question: 11
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 $\mathbf{E}\left[(X+Y-t)^{2}\right]$, when $t$ varies over all real numbers? $7$ $5$ $1.5$ $3.5$ $2.5$

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}\...

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112

TIFR ECE 2020 | Question: 11
Suppose that $X$ is a real valued random variable and $E[\exp X]=2$. Then, which of the following must be $\text{TRUE? Hint:}$ $(\exp (x)+\exp (y)) / 2 \geq \exp ((x+y) / 2)$. $E[X]<\ln 2$ $E[X]>\ln 2$ $E[X] \geq \ln 2$ $E[X] \leq \ln 2$ None of the above

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) /...

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113

TIFR ECE 2019 | Question: 9
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 sides of the coin. What is the expected number of times we would have seen tails? (Hint: the expected number of ... $(1/p.)$ $\frac{1}{p}$ $1+\frac{1}{1-p}$ $p+\frac{1}{p}-1$ $2$ None of the above

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...

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114

TIFR ECE 2019 | Question: 15
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 function (p.d.f.) $f$ where \[f(x)=\frac{1}{10} \exp (-x / 10)\] for $x \geq 0$, ... time, measured in minutes after $\text{9:00 AM,}$ would Anu expect the bus to arrive? $12.5$ $15$ $7.5$ $10$ $12.5$

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...

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115

TIFR ECE 2019 | Question: 10
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 $Z_{1}$ and $Z_{2}$ are such that if we define $Y_{1}=X+Z_{1}$ and $Y_{2}=X+Z_{2}$, where addition ... $\left(1 / p_{1}+1 / p_{2}\right)^{-1}$ $\left(1+1 / p_{1}+1 / p_{2}\right)^{-1}$ None of the above

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_{...

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116

TIFR ECE 2019 | Question: 6
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 equal probability. What is the value of the conditional expectation \[\mathbf{E}\left[\max \left(X_{1}, X_{2}\right) \mid \min \left(X_{1}, X_{2}\right)=3\right] ?\] $33 / 7$ $4$ $5$ $9 / 2$ $19 / 4$

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...

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117

TIFR ECE 2019 | Question: 12
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 the expected total number of balls drawn by this process? (Hint: Consider deriving an appropriate recurrence.) $\frac{a+b}{a+1}$ $\frac{a+b+1}{a}$ $\frac{a+b}{a}$ $\frac{a+b+1}{a+1}$ $a$

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...

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118

TIFR ECE 2016 | Question: 11
Suppose that a random variable $X$ has a probability density function (pdf) given by \[f(x)=c \exp (-2 x)\] for $x \geq 1$, and $f(x)=0$, for $x<1$, where $c$ is an appropriate constant so that $f(x)$ is a valid pdf. What is the expected value of $X$ given that $X \geq 5?$ $5 \frac{1}{2}$ $7$ $10$ $8 \frac{1}{2}$ $6$

Suppose that a random variable $X$ has a probability density function (pdf) given by\[f(x)=c \exp (-2 x)\]for $x \geq 1$, and $f(x)=0$, for $x<1$, where $c$ is an appropr...

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119

TIFR ECE 2019 | Question: 7
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 $p_{X}$ and $p_{Y}$, respectively, be the marginal p.m.f.'s of $X$ and $Y$, respectively. Which of ... None of the above

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...

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120

TIFR ECE 2016 | Question: 12
Recall that the Shannon entropy of a random variables $X$ taking values in a finite set $S$ is given by \[H[X]=-\sum_{x \in S} \operatorname{Pr}[X=x] \log _{2} \operatorname{Pr}[X=x] .\] (We set $0 \log _{2} 0=0$.) For a pair of random variables $(X, Y)$ taking ... $H\left[R_{513}, C_{513} \mid R_{1}, R_{2}, \ldots, R_{512}\right]?$ $\log _{2} 513$ $9$ $10$ $19$ $81$

Recall that the Shannon entropy of a random variables $X$ taking values in a finite set $S$ is given by\[H[X]=-\sum_{x \in S} \operatorname{Pr}[X=x] \log _{2} \operatorna...

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