GO Electronics - Recent questions in Probability and Statistics
https://ec.gateoverflow.in/questions/engineering-mathematics/probability-and-statistics
Powered by Question2AnswerGATE ECE 2021 | Question: 3
https://ec.gateoverflow.in/1610/gate-ece-2021-question-3
<p>Two continuous random variables $X$ and $Y$ are related as</p>
<p>$$Y=2X+3$$</p>
<p>Let $\sigma ^{2}_{X}$ and $\sigma ^{2}_{Y}$denote the variances of $X$ and $Y$, respectively. The variances are related as</p>
<ol style="list-style-type:upper-alpha" type="A">
<li>$\sigma ^{2}_{Y}=2 \sigma ^{2}_{X}$</li>
<li>$\sigma ^{2}_{Y}=4 \sigma ^{2}_{X}$</li>
<li>$\sigma ^{2}_{Y}=5 \sigma ^{2}_{X}$</li>
<li>$\sigma ^{2}_{Y}=25 \sigma ^{2}_{X}$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/1610/gate-ece-2021-question-3Fri, 19 Feb 2021 22:13:49 +0000GATE ECE 2021 | Question: 27
https://ec.gateoverflow.in/1586/gate-ece-2021-question-27
<p>A box contains the following three coins.</p>
<ol start="1" style="list-style-type:upper-roman">
<li>A fair coin with head on one face and tail on the other face.</li>
<li>A coin with heads on both the faces.</li>
<li>A coin with tails on both the faces.</li>
</ol>
<p>A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is</p>
<ol style="list-style-type:upper-alpha" type="A">
<li>$\frac{2}{5}$</li>
<li>$\frac{1}{3}$</li>
<li>$\frac{1}{2}$</li>
<li>$\frac{2}{3}$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/1586/gate-ece-2021-question-27Fri, 19 Feb 2021 22:13:46 +0000GATE ECE 2020 | Question: 25
https://ec.gateoverflow.in/1498/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 ___________.Probability and Statisticshttps://ec.gateoverflow.in/1498/gate-ece-2020-question-25Thu, 13 Feb 2020 07:29:11 +0000GATE ECE 2020 | Question: 54
https://ec.gateoverflow.in/1469/gate-ece-2020-question-54
$X$ is a random variable with uniform probability density function in the interval $[-2,\:10]$. For $Y=2X-6$, the conditional probability $P\left ( Y\leq 7\mid X\geq 5 \right )$ (rounded off to three decimal places) is __________.Probability and Statisticshttps://ec.gateoverflow.in/1469/gate-ece-2020-question-54Thu, 13 Feb 2020 07:29:07 +0000GATE2016 EC-3: 3
https://ec.gateoverflow.in/1460/gate2016-ec-3-3
The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is _________Probability and Statisticshttps://ec.gateoverflow.in/1460/gate2016-ec-3-3Thu, 21 Nov 2019 10:25:27 +0000GATE2009 EC: 11
https://ec.gateoverflow.in/1459/gate2009-ec-11
A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads.<br />
<br />
(A) $\left(\dfrac{1}{2}\right)^{2}$<br />
<br />
(B) $^{10}C_2\left(\dfrac{1}{2}\right)^{2}$<br />
<br />
(C) $\left(\dfrac{1}{2}\right)^{10}$<br />
<br />
(D) $^{10}C_2\left(\dfrac{1}{2}\right)^{10}$Probability and Statisticshttps://ec.gateoverflow.in/1459/gate2009-ec-11Thu, 21 Nov 2019 09:40:30 +0000GATE ECE 2019 | Question: 18
https://ec.gateoverflow.in/1383/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 $E\left[X\right]+E\left[Y\right]=$___________.Probability and Statisticshttps://ec.gateoverflow.in/1383/gate-ece-2019-question-18Tue, 12 Feb 2019 17:11:06 +0000GATE ECE 2019 | Question: 20
https://ec.gateoverflow.in/1381/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<br />
<br />
$$F_{Z}(x)= \left\{\begin{matrix} 1-e^{-x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$$<br />
<br />
Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.Probability and Statisticshttps://ec.gateoverflow.in/1381/gate-ece-2019-question-20Tue, 12 Feb 2019 17:11:06 +0000GATE ECE 2019 | Question: 47
https://ec.gateoverflow.in/1354/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 independent of $X,$ and is uniformly distributed over the interval $[-2,2].$ The receiver makes a decision<br />
<br />
$$\hat{X}= \left\{\begin{matrix}<br />
-1,& \text{if} \quad Y \leq \theta \\<br />
+1,& \text{if} \quad Y > \theta<br />
\end{matrix}\right. $$<br />
where the threshold $\theta \in [-1,1]$ is chosen so as to minimize the probability of error $Pr[ \hat{X} \neq X].$ The minimum probability of error, rounded off to $1$ decimal place, is _________.Probability and Statisticshttps://ec.gateoverflow.in/1354/gate-ece-2019-question-47Tue, 12 Feb 2019 17:11:03 +0000GATE ECE 2016 Set 3 | Question: 3
https://ec.gateoverflow.in/991/gate-ece-2016-set-3-question-3
The probability of getting a “head” in a single toss of a biased coin is $0.3$. The coin is tossed repeatedly till a head is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is _______Probability and Statisticshttps://ec.gateoverflow.in/991/gate-ece-2016-set-3-question-3Tue, 27 Mar 2018 20:33:58 +0000GATE ECE 2016 Set 3 | Question: 51
https://ec.gateoverflow.in/943/gate-ece-2016-set-3-question-51
The bit error probability of a memoryless binary symmetric channel is $10^{-5}$. If $10^5$ bits are sent over this channel, then the probability that not more than one bit will be in error is _______Probability and Statisticshttps://ec.gateoverflow.in/943/gate-ece-2016-set-3-question-51Tue, 27 Mar 2018 20:33:50 +0000GATE ECE 2016 Set 2 | Question: 21
https://ec.gateoverflow.in/908/gate-ece-2016-set-2-question-21
A discrete memoryless source has an alphabet $\left \{ a_{1},a_{2}, a_{3},a_{4}\right \}$ with corresponding probabilities $\left \{ \frac{1}{2}, \frac{1}{4},\frac{1}{8},\frac{1}{8}\right \}.$ The minimum required average codeword length in bits to represent this source for error-free reconstruction is _________Probability and Statisticshttps://ec.gateoverflow.in/908/gate-ece-2016-set-2-question-21Tue, 27 Mar 2018 20:23:28 +0000GATE ECE 2016 Set 2 | Question: 28
https://ec.gateoverflow.in/901/gate-ece-2016-set-2-question-28
Two random variables $X$ and $Y$ are distributed according to $$f_{X,Y}(x,y)=\begin{cases} (x+y),& 0\leq x\leq 1,&0\leq y\leq 1\\ 0, & \text{otherwise.} \end{cases}$$ The probability $P(X+Y\leq 1)$ is ________Probability and Statisticshttps://ec.gateoverflow.in/901/gate-ece-2016-set-2-question-28Tue, 27 Mar 2018 20:23:27 +0000GATE ECE 2016 Set 1 | Question: 2
https://ec.gateoverflow.in/862/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 _____Probability and Statisticshttps://ec.gateoverflow.in/862/gate-ece-2016-set-1-question-2Tue, 27 Mar 2018 20:09:24 +0000GATE ECE 2016 Set 1 | Question: 48
https://ec.gateoverflow.in/816/gate-ece-2016-set-1-question-48
Consider a discrete memoryless source with alphabet $S = \{s_0,s_1,s_2,s_3,s_4, \dots \}$ and respective probabilities of occurence $P = \bigg\{ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots \bigg\}$. The entropy of the source (in bits) is _________Probability and Statisticshttps://ec.gateoverflow.in/816/gate-ece-2016-set-1-question-48Tue, 27 Mar 2018 20:09:17 +0000GATE ECE 2015 Set 3 | Question: 27
https://ec.gateoverflow.in/772/gate-ece-2015-set-3-question-27
A fair die with faces $\{1, 2, 3, 4, 5, 6\}$ is thrown repeatedly till $’3’$ is observed for the first time. Let $X$ denote the number of times the die is thrown. The expected value of $X$ is _______.Probability and Statisticshttps://ec.gateoverflow.in/772/gate-ece-2015-set-3-question-27Tue, 27 Mar 2018 19:53:25 +0000GATE ECE 2015 Set 3 | Question: 50
https://ec.gateoverflow.in/749/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 __________.Probability and Statisticshttps://ec.gateoverflow.in/749/gate-ece-2015-set-3-question-50Tue, 27 Mar 2018 19:53:22 +0000GATE ECE 2015 Set 3 | Question: 52
https://ec.gateoverflow.in/747/gate-ece-2015-set-3-question-52
A random binary wave $y(t)$ is given by<br />
<br />
$$y(t) = \sum_{n = -\infty}^{\infty}X_{n}\:p(t-nT-\phi)$$<br />
<br />
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 $[0,T].$The sequence $\{X_{n}\}$ consist of independent and identically distributed binary valued random variables with $P\{X_{n} = +1\} = P\{X_{n} = -1\} = 0.5$ for each $n.$<br />
<br />
The value of the autocorrelation $R_{yy}\left(\dfrac{3T}{4}\right) \underset{=}{\Delta} E\left[y(t)y\left(t-\dfrac{3T}{4}\right)\right]$ equals _________.Probability and Statisticshttps://ec.gateoverflow.in/747/gate-ece-2015-set-3-question-52Tue, 27 Mar 2018 19:53:21 +0000GATE ECE 2015 Set 2 | Question: 29
https://ec.gateoverflow.in/705/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 _______.Probability and Statisticshttps://ec.gateoverflow.in/705/gate-ece-2015-set-2-question-29Tue, 27 Mar 2018 19:52:05 +0000GATE ECE 2015 Set 2 | Question: 52
https://ec.gateoverflow.in/682/gate-ece-2015-set-2-question-52
<p>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</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$pq+(1-p)(1-q)$</li>
<li>$pq$</li>
<li>$p(1-q)$</li>
<li>$1-pq$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/682/gate-ece-2015-set-2-question-52Tue, 27 Mar 2018 19:52:02 +0000GATE ECE 2015 Set 1 | Question: 3
https://ec.gateoverflow.in/666/gate-ece-2015-set-1-question-3
<p>Suppose $A$ and $B$ are two independent events with probabilities $P(A) \neq 0$ and $P(B) \neq 0$. Let $\overline{A}$ and $\overline{B}$ be their complements. Which one of the following statements is FALSE?</p>
<ol style="list-style-type:upper-alpha" type="A">
<li>$P(A \cap B) = P(A)P(B)$</li>
<li>$P(A \mid B) = P(A)$</li>
<li>$P(A \cup B) = P(A) + P(B)$</li>
<li>$P(\overline{A} \cap \overline{B} )= P(\overline{A})P(\overline{B})$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/666/gate-ece-2015-set-1-question-3Tue, 27 Mar 2018 19:50:43 +0000GATE ECE 2015 Set 1 | Question: 49
https://ec.gateoverflow.in/620/gate-ece-2015-set-1-question-49
<p>The input $X$ to the Binary Symmetric Channel (BSC) shown in the figure is $’1’$ with probability $0.8$. The cross-over probability is $1/7$. If the received bit $Y=0$, the conditional probability that $’1’$ was transmitted is ____________</p>
<p><img alt="" src="https://ec.gateoverflow.in/?qa=blob&qa_blobid=3981316333587993271"></p>Probability and Statisticshttps://ec.gateoverflow.in/620/gate-ece-2015-set-1-question-49Tue, 27 Mar 2018 19:50:35 +0000GATE ECE 2015 Set 1 | Question: 52
https://ec.gateoverflow.in/617/gate-ece-2015-set-1-question-52
<p>A source emits bit $0$ with probability $\frac{1}{3}$ and bit $1$ with probability $\frac{2}{3}$. The emitted bits are communicated to the receiver. The receiver decides for either $0$ or $1$ based on the received value $R$. It is given that the conditional density functions of $R$ are as $$f_{R \mid 0}( r) = \begin{cases} \frac{1}{4}, & -3 \leq x \leq 1, \\ 0, & \text{otherwise,} \end{cases} \text{ and } f_{R \mid 1}( r) = \begin{cases} \frac{1}{6}, & -1 \leq x \leq 5, \\ 0, & \text{otherwise,} \end{cases}$$ </p>
<p>The minimum decision error probability is</p>
<ol style="list-style-type:upper-alpha" type="A">
<li>$0$</li>
<li>$1/12$</li>
<li>$1/9$</li>
<li>$1/6$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/617/gate-ece-2015-set-1-question-52Tue, 27 Mar 2018 19:50:34 +0000GATE ECE 2014 Set 4 | Question: 27
https://ec.gateoverflow.in/577/gate-ece-2014-set-4-question-27
Parcels from sender S to receiver R pass sequentially through two-post offices. Each post-office has a probability $\frac{1}{5}$ of losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post office is _________Probability and Statisticshttps://ec.gateoverflow.in/577/gate-ece-2014-set-4-question-27Mon, 26 Mar 2018 00:35:14 +0000GATE ECE 2014 Set 4 | Question: 50
https://ec.gateoverflow.in/554/gate-ece-2014-set-4-question-50
<p>Consider the $Z$-channel given in the figure. The input is $0$ or $1$ with equal probability.</p>
<p><img alt="" src="https://ec.gateoverflow.in/?qa=blob&qa_blobid=18352383822187495364"></p>
<p>If the output is $0$, the probability that the input is also $0$ equals ___________</p>Probability and Statisticshttps://ec.gateoverflow.in/554/gate-ece-2014-set-4-question-50Mon, 26 Mar 2018 00:35:11 +0000GATE ECE 2014 Set 3 | Question: 4
https://ec.gateoverflow.in/535/gate-ece-2014-set-3-question-4
<p>An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$0.067$</li>
<li>$0.073$</li>
<li>$0.082$</li>
<li>$0.091$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/535/gate-ece-2014-set-3-question-4Mon, 26 Mar 2018 00:20:12 +0000GATE ECE 2014 Set 3 | Question: 28
https://ec.gateoverflow.in/511/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 _______ .Probability and Statisticshttps://ec.gateoverflow.in/511/gate-ece-2014-set-3-question-28Mon, 26 Mar 2018 00:20:09 +0000GATE ECE 2014 Set 3 | Question: 29
https://ec.gateoverflow.in/510/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 _________.Probability and Statisticshttps://ec.gateoverflow.in/510/gate-ece-2014-set-3-question-29Mon, 26 Mar 2018 00:20:09 +0000GATE ECE 2014 Set 3 | Question: 52
https://ec.gateoverflow.in/487/gate-ece-2014-set-3-question-52
<p>A binary random variable $X$ takes the value of $1$ with probability $1/3$. $X$ is input to a cascade of $2$ independent identical binary symmetric channels (BSCs) each with crossover probability $1/2$. The output of BSCs are the random variables $Y_{1}$ and $Y_{2}$ as shown in the figure.</p>
<p> <img alt="" src="https://ec.gateoverflow.in/?qa=blob&qa_blobid=16400646578829208576"></p>
<p>The value of $H( Y_{1} )+H( Y_{2} )$ in bits is ______.</p>Probability and Statisticshttps://ec.gateoverflow.in/487/gate-ece-2014-set-3-question-52Mon, 26 Mar 2018 00:20:06 +0000GATE ECE 2014 Set 2 | Question: 2
https://ec.gateoverflow.in/472/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 ________.Probability and Statisticshttps://ec.gateoverflow.in/472/gate-ece-2014-set-2-question-2Mon, 26 Mar 2018 00:04:39 +0000GATE ECE 2014 Set 2 | Question: 49
https://ec.gateoverflow.in/425/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 ______.Probability and Statisticshttps://ec.gateoverflow.in/425/gate-ece-2014-set-2-question-49Mon, 26 Mar 2018 00:04:33 +0000GATE ECE 2014 Set 1 | Question: 2
https://ec.gateoverflow.in/407/gate-ece-2014-set-1-question-2
In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is ________.Probability and Statisticshttps://ec.gateoverflow.in/407/gate-ece-2014-set-1-question-2Sun, 25 Mar 2018 23:47:36 +0000GATE ECE 2014 Set 1 | Question: 5
https://ec.gateoverflow.in/404/gate-ece-2014-set-1-question-5
Let $X_{1},X_{2},$ and $X_{3}$ be independent and identically distributed random variables with the uniform distribution on $[0,1].$ The probability $P\{X_{1}\: \text{is the largest}\}$ is ________.Probability and Statisticshttps://ec.gateoverflow.in/404/gate-ece-2014-set-1-question-5Sun, 25 Mar 2018 23:47:36 +0000GATE ECE 2014 Set 1 | Question: 49
https://ec.gateoverflow.in/360/gate-ece-2014-set-1-question-49
<p>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</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$(E[X])^{2}>E[X^{2}]$</li>
<li>$E[X^{2}]\geq (E[X])^{2}$</li>
<li>$E[X^{2}] = (E[X])^{2}$</li>
<li>$E[X^{2}] > (E[X])^{2}$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/360/gate-ece-2014-set-1-question-49Sun, 25 Mar 2018 23:47:29 +0000GATE ECE 2014 Set 1 | Question: 50
https://ec.gateoverflow.in/359/gate-ece-2014-set-1-question-50
<p>Consider a random process $X(t) = \sqrt{2}\sin(2\pi t + \varphi),$ where the random phase $\varphi$ is uniformly distributed in the interval $[0,2\pi].$ The auto-correlation $E[X(t_{1})X(t_{2})]$ is</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$\cos(2\pi(t_{1} + t_{2}))$</li>
<li>$\sin(2\pi(t_{1} - t_{2}))$</li>
<li>$\sin(2\pi(t_{1} + t_{2}))$</li>
<li>$\cos(2\pi(t_{1} - t_{2}))$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/359/gate-ece-2014-set-1-question-50Sun, 25 Mar 2018 23:47:29 +0000GATE ECE 2013 | Question: 38
https://ec.gateoverflow.in/326/gate-ece-2013-question-38
<p>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 $x$</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$F(x) - G(x) \leq 0$</li>
<li>$F(x) - G(x) \geq 0$</li>
<li>$(F(x) - G(x)) \cdot x\leq 0$</li>
<li>$(F(x) - G(x)) \cdot x\geq 0$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/326/gate-ece-2013-question-38Sun, 25 Mar 2018 21:00:41 +0000GATE ECE 2013 | Question: 26
https://ec.gateoverflow.in/314/gate-ece-2013-question-26
<p>Let $U$ and $V$ be two independent zero mean Gaussian random variables of variances $\dfrac{1}{4}$ and $\dfrac{1}{9}$ respectively. The probability $P(3V\geq 2U)$ is</p>
<ol start="1" style="list-style-type:upper-alpha">
<li>$4/9$</li>
<li>$1/2$</li>
<li>$2/3$</li>
<li>$5/9$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/314/gate-ece-2013-question-26Sun, 25 Mar 2018 21:00:39 +0000GATE ECE 2012 | Question: 38
https://ec.gateoverflow.in/261/gate-ece-2012-question-38
<p>A binary symmetric channel (BSC) has a transition probability of $\frac{1}{8}$. If the binary transmit symbol $X$ is such that $P(X=0)\:=\:\frac{9}{10}$, then the probability of error for an optimum receiver will be</p>
<ol style="list-style-type:upper-alpha" type="A">
<li>$\frac{7}{80}$</li>
<li>$\frac{63}{80}$</li>
<li>$\frac{9}{10}$</li>
<li>$\frac{1}{10}$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/261/gate-ece-2012-question-38Sun, 25 Mar 2018 09:55:09 +0000GATE ECE 2012 | Question: 36
https://ec.gateoverflow.in/259/gate-ece-2012-question-36
<p>A fair coin is tossed till head appears for the first time. The probability that the number of required tosses is odd, is</p>
<ol style="list-style-type:upper-alpha" type="A">
<li>$\frac{1}{3}$</li>
<li>$\frac{1}{2}$</li>
<li>$\frac{2}{3}$</li>
<li>$\frac{3}{4}$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/259/gate-ece-2012-question-36Sun, 25 Mar 2018 09:55:09 +0000GATE ECE 2012 | Question: 24
https://ec.gateoverflow.in/247/gate-ece-2012-question-24
<p>Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[-1,1]$. The probability that max$[X,Y]$ is less than $\frac{1}{2}$ is</p>
<ol style="list-style-type:upper-alpha" type="A">
<li>$\frac{3}{4}$</li>
<li>$\frac{9}{16}$</li>
<li>$\frac{1}{4}$</li>
<li>$\frac{2}{3}$</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/247/gate-ece-2012-question-24Sun, 25 Mar 2018 09:55:07 +0000GATE ECE 2012 | Question: 15
https://ec.gateoverflow.in/238/gate-ece-2012-question-15
<p>A source alphabet consists of $N$ symbols with the probability of the first two symbols being the same. A source encoder increases the probability of the first symbol by a small amount $\varepsilon$ and decreases that of the second by $\varepsilon$. After encoding, the entropy of the source</p>
<ol style="list-style-type:upper-alpha" type="A">
<li>increases</li>
<li>remains the same</li>
<li>increases only if $N=2$</li>
<li>decreases</li>
</ol>Probability and Statisticshttps://ec.gateoverflow.in/238/gate-ece-2012-question-15Sun, 25 Mar 2018 09:55:06 +0000GATE ECE 2018 | Question: 40
https://ec.gateoverflow.in/170/gate-ece-2018-question-40
A random variable $X$ takes values $-0.5$ and $0.5$ with probabilities $\dfrac{1}{4}$ and $\dfrac{3}{4}$, respectively. The noisy observation of $X\:\text{is}\:Y=X+Z,$ where $Z$ has uniform probability density over the interval $(-1,1).\: X$ and $Z$ are independent. If the MAP rule based detector outputs $\hat{X}$ as<br />
$$\hat{X}=\left\{\begin{matrix} -0.5, & Y<\alpha \\ 0.5,& Y\geq \alpha, \end{matrix}\right.$$ then the values of $\alpha$ (accurate to two decimal places) is ________.Probability and Statisticshttps://ec.gateoverflow.in/170/gate-ece-2018-question-40Mon, 19 Feb 2018 04:14:04 +0000GATE ECE 2018 | Question: 23
https://ec.gateoverflow.in/153/gate-ece-2018-question-23
Let $X_{1},\:X_{2},\:X_{3}$ and $X_{4}$ be independent normal random variable with zero mean and unit variance. The probability that $X_{4}$ is the smallest among the four is ________.Probability and Statisticshttps://ec.gateoverflow.in/153/gate-ece-2018-question-23Mon, 19 Feb 2018 04:14:02 +0000GATE ECE 2017 Set 2 | Question: 29
https://ec.gateoverflow.in/104/gate-ece-2017-set-2-question-29
Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is $40 \%$ chance of getting reservation in any attempt by a passenger, then the average number of attempts that passengers need to make to get a seat reserved is __________Probability and Statisticshttps://ec.gateoverflow.in/104/gate-ece-2017-set-2-question-29Thu, 23 Nov 2017 07:44:14 +0000GATE ECE 2017 Set 2 | Question: 22
https://ec.gateoverflow.in/97/gate-ece-2017-set-2-question-22
Consider the random process<br />
<br />
$X(t)=U+Vt,$<br />
<br />
Where $U$ is a zero-mean Gaussian random variable and V is a random variable uniformly distributed between $0$ and $2$. Assume that $U$ and $V$ are statistically independent. The mean value of the random process at $t = 2$ is ________Probability and Statisticshttps://ec.gateoverflow.in/97/gate-ece-2017-set-2-question-22Thu, 23 Nov 2017 07:44:13 +0000GATE ECE 2017 Set 1 | Question: 4
https://ec.gateoverflow.in/14/gate-ece-2017-set-1-question-4
Three fair cubical dice are thrown simultaneously . The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place)________.Probability and Statisticshttps://ec.gateoverflow.in/14/gate-ece-2017-set-1-question-4Fri, 17 Nov 2017 04:16:54 +0000