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Recent questions tagged random-variable
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TIFR ECE 2015 | Question: 8
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 independent $E(X Z)=E(Y W)$ $E(X Y)=E(Z W)$ $(a), (b)$, and $(c)$ $(a)$ and $(b)$ only
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TIFR ECE 2015 | Question: 9
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 variance of $Y?$ $121 / 4$ $37 / 20 $ $9 / 5$ $99 / 12$ None of the above
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TIFR ECE 2015 | Question: 10
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? $E(M)=\infty$ $E(M) \in[5,10]$ $E(M)=\exp (1)$ $E(M)=\pi$ None of the above
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4
TIFR ECE 2014 | Question: 3
For a non-negative continuous random variable $X$, which of the following is TRUE? $E\{X\}=\int_{0}^{\infty} P(X>x) d x$. $E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$. $P(X<x) \leq \frac{E\{X\}}{x}$. $(a)$ and $(c)$. None of the above.
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5
TIFR ECE 2014 | Question: 12
Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal value of $\alpha$ which minimizes $\mathbb{E}\left[(X-\alpha Y)^{2}\right]$ ... $1$ $\frac{\sigma_{Y}^{2}}{\sigma_{Z}^{2}}$ None of the above.
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6
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 $(b)$
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7
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
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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
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9
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? $1 / 2$ $3 / 8$ $3 / 5$ $3 / 4$ $5 / 7$
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10
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 $n \rightarrow \infty$
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TIFR ECE 2022 | Question: 11
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 of the square of the distance after $100$ seconds from the initial position? Hint: ... sum of independent and identically distributed random variables. $100$ $\frac{\sqrt{300}}{4}$ $40$ $200$ $20 \pi$
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12
TIFR ECE 2021 | Question: 11
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 outcome. Further, $U$ denotes a continuous random variable that is independent of $X_{1}$ and $X_{2}$ ... on this sum what is the probability that $X_{1}$ equals $2?$ $2.21$ $3$ $1 / 6$ $1 / 5$ $1 / 3$
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13
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 ... $\frac{10}{36}$ $\frac{11}{36}$ $\frac{12}{36}$ $\frac{13}{36}$ $\frac{14}{36}$
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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
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15
TIFR ECE 2020 | Question: 7
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 $E\left[\left(z_{1}+\right.\right.$ $\left.\left.\ldots z_{n}\right)^{2}\right] ?$ $0$ $n p+n(n-1) p^{2}$ $n^{3} p^{2}$ $n^{2} p^{2}+n p$ None of the above
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16
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
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17
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 the ... None of the above
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18
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
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19
TIFR ECE 2018 | Question: 9
Let $X$ and $Y$ be two independent and identically distributed binary random variables that take values $\{-1,+1\}$ each with probability $1 / 2$. Let $Z_{1}=\max (X, Y)$, and $Z_{2}=\min (X, Y)$. Consider the following statements. $Z_{1}$ and $Z_{2}$ are uncorrelated ... $\text{(iii)}$ Both $\text{(i) and (ii), but not (iii)}$ All of $\text{(i), (ii) and (iii)}$
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20
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
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21
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 $Y=\max \left\{X_{1}, X_{2}\right\}$ ... $\operatorname{Pr}[Z=1]>\operatorname{Pr}[Z=2]=\frac{1}{2}$ $\operatorname{Pr}[Z=1]<\operatorname{Pr}[Z=2]$
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22
TIFR ECE 2016 | Question: 3
Let $(X, Y)$ be a pair of independent random variables. Suppose $X$ takes values in $\{1, \ldots, 6\}$ with equal probability, and $Y$ takes values in $\{2,3\}$ with $\operatorname{Pr}[Y=2]=p$. Let $Z=(X \bmod Y)+1$ ... $\operatorname{Pr}[Z=1]=\frac{1}{2}$ for $p=\frac{1}{2}$ $\operatorname{Pr}[Z=1]=p(1-p)$ None of the above
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23
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 ... $1$ $\frac{2}{3}$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$
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24
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 ... $H\left[R_{513}, C_{513} \mid R_{1}, R_{2}, \ldots, R_{512}\right]?$ $\log _{2} 513$ $9$ $10$ $19$ $81$
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25
GATE ECE 2021 | Question: 3
Two continuous random variables $X$ and $Y$ are related as $Y=2X+3$ Let $\sigma ^{2}_{X}$ and $\sigma ^{2}_{Y}$denote the variances of $X$ and $Y$, respectively. The variances are related as $\sigma ^{2}_{Y}=2 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=4 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=5 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=25 \sigma ^{2}_{X}$
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GATE ECE 2020 | Question: 25
The two sides of a fair coin are labelled as $0$ to $1$. The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X = \text{min}(M, N)$, the expected value $E(X)$ (rounded off to two decimal places) is ___________.
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27
GATE ECE 2019 | Question: 18
If $X$ and $Y$ are random variables such that $E\left[2X+Y\right]=0$ and $E\left[X+2Y\right]=33$, then $E\left[X\right]+E\left[Y\right]=$___________.
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28
GATE ECE 2019 | Question: 20
Let $Z$ be an exponential random variable with mean $1$. That is, the cumulative distribution function of $Z$ is given by $F_{Z}(x)= \left\{\begin{matrix} 1-e^{-x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$ Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.
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gate2019-ec
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29
GATE ECE 2019 | Question: 47
A random variable $X$ takes values $-1$ and $+1$ with probabilities $0.2$ and $0.8$, respectively. It is transmitted across a channel which adds noise $N,$ so that the random variable at the channel output is $Y=X+N$. The noise $N$ is ... probability of error $Pr[ \hat{X} \neq X].$ The minimum probability of error, rounded off to $1$ decimal place, is _________.
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30
GATE ECE 2016 Set 1 | Question: 2
The second moment of a Poisson-distributed random variable is $2$. The mean of the random variable is _____
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31
GATE ECE 2015 Set 3 | Question: 50
The variance of the random variable $X$ with probability density function $f(x)=\dfrac{1}{2}\mid x \mid e^{- \mid x \mid}$ is __________.
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32
GATE ECE 2015 Set 3 | Question: 52
A random binary wave $y(t)$ is given by $y(t) = \sum_{n = -\infty}^{\infty}X_{n}\:p(t-nT-\phi)$ where $p(t)=u(t)-u(t-T),u(t)$ is the unit step function and $\phi$ is an independent random variable with uniform distribution in $[0,T].$ ... $R_{yy}\left(\dfrac{3T}{4}\right) \underset{=}{\Delta} E\left[y(t)y\left(t-\dfrac{3T}{4}\right)\right]$ equals _________.
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33
GATE ECE 2015 Set 2 | Question: 29
Let the random variable $X$ represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of $X$ is _______.
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34
GATE ECE 2015 Set 2 | Question: 52
Let $X\in \{0,1\}$ and $Y\in \{0,1\}$ be two independent binary random variables. If $P(X=0)=p$ and $P(Y=0)=q,$ then $P(X+Y\geq 1)$ is equal to $pq+(1-p)(1-q)$ $pq$ $p(1-q)$ $1-pq$
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35
GATE ECE 2014 Set 4 | Question: 3
Let $X$ be a zero mean unit variance Gaussian random variable. $E[ \mid X \mid ]$ is equal to __________
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36
GATE ECE 2014 Set 3 | Question: 29
Let $X_{1},X_{2},$ and $X_{3}$ be independent and identically distributed random variables with the uniform distribution on $[0,1]$. The probability $P\left \{ X_{1}+X_{2}\leq X_{3}\right \}$ is _________.
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37
GATE ECE 2014 Set 2 | Question: 2
Let $X$ be a random variable which is uniformly chosen from the set of positive odd numbers less than $100$. The expectation, $E[X]$, is ________.
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38
GATE ECE 2014 Set 2 | Question: 49
The input to a $1$ – bit quantizer is a random variable $X$ with pdf $f_{X}( x )= 2e^{-2x}$ for $x\geq 0$ and $f_{X} (x )= 0$ for $x< 0$. For outputs to be of equal probability, the quantizer threshold should be ______.
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39
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 $(E[X])^{2}>E[X^{2}]$ $E[X^{2}]\geq (E[X])^{2}$ $E[X^{2}] = (E[X])^{2}$ $E[X^{2}] > (E[X])^{2}$
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40
GATE ECE 2013 | Question: 38
Consider two identically distributed zero-mean random variables $U$ and $V.$ Let the cumulative distribution functions of $U$ and $2V$ be $F(x)$ and $G(x)$ respectively. Then, for all values of $x$ $F(x) - G(x) \leq 0$ $F(x) - G(x) \geq 0$ $(F(x) - G(x)) \cdot x\leq 0$ $(F(x) - G(x)) \cdot x\geq 0$
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