Recent questions and answers in Probability and Statistics - GO Electronics

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1

GATE2014-1-2
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 ________.

answered Oct 5 in Probability and Statistics by Gyanu (140 points)

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2

GATE2020-EC: 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 __________.

asked Feb 13 in Probability and Statistics by jothee (1.4k points)

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3

GATE2012-36
A fair coin is tossed till head appears for the first time. The probability that the number of required tosses is odd, is $\frac{1}{3}$ $\frac{1}{2}$ $\frac{2}{3}$ $\frac{3}{4}$

answered Nov 26, 2019 in Probability and Statistics by Naveen Kumar 3 (150 points)

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4

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 _________

asked Nov 21, 2019 in Probability and Statistics by KUSHAGRA गुप्ता (260 points)

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5

GATE2009 EC: 11
A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads. (A) $\left(\dfrac{1}{2}\right)^{2}$ (B) $^{10}C_2\left(\dfrac{1}{2}\right)^{2}$ (C) $\left(\dfrac{1}{2}\right)^{10}$ (D) $^{10}C_2\left(\dfrac{1}{2}\right)^{10}$

asked Nov 21, 2019 in Probability and Statistics by KUSHAGRA गुप्ता (260 points)

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6

GATE2019 EC: 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 ... the probability of error $Pr[ \hat{X} \neq X].$ The minimum probability of error, rounded off to $1$ decimal place, is _________.

asked Feb 12, 2019 in Probability and Statistics by Arjun (1.4k points)

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7

GATE2016-3-51
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 _______

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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8

GATE2016-2-21
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 _________

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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9

GATE2016-2-28
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 ________

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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10

GATE2016-1-2
The second moment of a Poisson-distributed random variable is $2$. The mean of the random variable is _____

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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11

GATE2016-1-48
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 _________

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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12

GATE2015-3-50
The variance of the random variable $X$ with probability density function $f(x)=\dfrac{1}{2}\mid x \mid e^{- \mid x \mid}$ is __________.

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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13

GATE2015-1-3
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? $P(A \cap B) = P(A)P(B)$ $P(A \mid B) = P(A)$ $P(A \cup B) = P(A) + P(B)$ $P(\overline{A} \cap \overline{B} )= P(\overline{A})P(\overline{B})$

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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14

GATE2015-1-49
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 ____________

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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15

GATE2015-1-52
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 ... $0$ $1/12$ $1/9$ $1/6$

asked Mar 28, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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16

GATE2014-4-1
The series $\Sigma_{n=0}^{\infty} \frac{1}{n!}$ converges to $2 \text{ ln } 2$ $\sqrt{2}$ $2$ $e$

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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17

GATE2014-4-27
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 _________

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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18

GATE2014-4-50
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 ___________

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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19

GATE2014-3-4
An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is $0.067$ $0.073$ $0.082$ $0.091$

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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20

GATE2014-3-29
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 _________.

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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21

GATE2014-3-52
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. The value of $H( Y_{1} )+H( Y_{2} )$ in bits is ______.

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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22

GATE2014-2-49
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 ______.

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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23

GATE2014-1-5
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 ________.

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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24

GATE2014-1-23
The capacity of a Binary Symmetric Channel $\text{(BSC)}$ with cross-over probability $0.5$ is ________.

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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25

GATE2014-1-49
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 $(E[X])^{2}>E[X^{2}]$ $E[X^{2}]\geq (E[X])^{2}$ $E[X^{2}] = (E[X])^{2}$ $E[X^{2}] > (E[X])^{2}$

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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26

GATE2014-1-50
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 $\cos(2\pi(t_{1} + t_{2}))$ $\sin(2\pi(t_{1} - t_{2}))$ $\sin(2\pi(t_{1} + t_{2}))$ $\cos(2\pi(t_{1} - t_{2}))$

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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27

GATE2013-38
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 $x$ $F(x) - G(x) \leq 0$ $F(x) - G(x) \geq 0$ $(F(x) - G(x)) \cdot x\leq 0$ $(F(x) - G(x)) \cdot x\geq 0$

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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28

GATE2013-26
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 $4/9$ $1/2$ $2/3$ $5/9$

asked Mar 26, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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29

GATE2012-38
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 $\frac{7}{80}$ $\frac{63}{80}$ $\frac{9}{10}$ $\frac{1}{10}$

asked Mar 25, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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30

GATE2012-24
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 $\frac{3}{4}$ $\frac{9}{16}$ $\frac{1}{4}$ $\frac{2}{3}$

asked Mar 25, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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31

GATE2012-15
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 increases remains the same increases only if $N=2$ decreases

asked Mar 25, 2018 in Probability and Statistics by Milicevic3306 (15.8k points)

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32

GATE2018-40
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 $Z$ ... $\alpha$ (accurate to two decimal places) is ________.

asked Feb 19, 2018 in Probability and Statistics by gatecse (1.4k points)

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33

GATE2018-23
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 ________.

asked Feb 19, 2018 in Probability and Statistics by gatecse (1.4k points)

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34

GATE2017 EC-2: 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 __________

asked Nov 23, 2017 in Probability and Statistics by admin (2.8k points)

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35

GATE2017 EC-2: 22
Consider the random process $X(t)=U+Vt,$ 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 ________

asked Nov 23, 2017 in Probability and Statistics by admin (2.8k points)

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36

GATE2017 EC-1: 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)________.

asked Nov 17, 2017 in Probability and Statistics by admin (2.8k points)

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