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Recent questions and answers in Probability and Statistics
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GATE EC 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 $\sigma ^{2}_{Y}=2 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=4 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=5 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=25 \sigma ^{2}_{X}$
asked
Feb 20
in
Probability and Statistics
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Arjun
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4.4k
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14
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2
GATE EC 2021  Question: 27
A box contains the following three coins. A fair coin with head on one face and tail on the other face. A coin with heads on both the faces. A coin with tails on both the faces. A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one ... getting a head in the second toss is $\frac{2}{5}$ $\frac{1}{3}$ $\frac{1}{2}$ $\frac{2}{3}$
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Feb 20
in
Probability and Statistics
by
Arjun
(
4.4k
points)

19
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gateec2021
probabilityandstatistics
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+1
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1
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3
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 ________.
answered
Oct 5, 2020
in
Probability and Statistics
by
Gyanu
(
140
points)

97
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gate2014ec1
numericalanswers
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0
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4
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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Feb 13, 2020
in
Probability and Statistics
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jothee
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1.8k
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73
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gate2020ec
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5
GATE ECE 2020  Question: 54
$X$ is a random variable with uniform probability density function in the interval $[2,\:10]$. For $Y=2X6$, the conditional probability $P\left ( Y\leq 7\mid X\geq 5 \right )$ (rounded off to three decimal places) is __________.
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Feb 13, 2020
in
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jothee
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1.8k
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23
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0
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6
GATE ECE 2012  Question: 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
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150
points)

28
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0
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7
GATE2016 EC3: 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 _________
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Nov 21, 2019
in
Probability and Statistics
by
KUSHAGRA गुप्ता
(
270
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27
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gate2016ec
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+1
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8
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) $^{10}C_2\left(\dfrac{1}{2}\right)^{10}$
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Nov 21, 2019
in
Probability and Statistics
by
KUSHAGRA गुप्ता
(
270
points)

18
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gate2009ec
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0
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0
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9
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 $E\left[X\right]+E\left[Y\right]=$___________.
asked
Feb 12, 2019
in
Probability and Statistics
by
Arjun
(
4.4k
points)

37
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gate2019ec
numericalanswers
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0
votes
0
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10
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 $F_{Z}(x)= \left\{\begin{matrix} 1e^{x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$ Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.
asked
Feb 12, 2019
in
Probability and Statistics
by
Arjun
(
4.4k
points)

75
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gate2019ec
numericalanswers
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0
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0
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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 _________.
asked
Feb 12, 2019
in
Probability and Statistics
by
Arjun
(
4.4k
points)

66
views
gate2019ec
numericalanswers
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uniformdistribution
0
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12
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 _______
asked
Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

40
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gate2016ec3
probabilityandstatistics
probability
independentevents
0
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0
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13
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 _______
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Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

22
views
gate2016ec3
numericalanswers
probabilityandstatistics
probability
0
votes
0
answers
14
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 errorfree reconstruction is _________
asked
Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

18
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gate2016ec2
numericalanswers
probabilityandstatistics
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0
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0
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15
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 ________
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Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
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15.8k
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25
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gate2016ec2
numericalanswers
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0
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16
GATE ECE 2016 Set 1  Question: 2
The second moment of a Poissondistributed random variable is $2$. The mean of the random variable is _____
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Mar 28, 2018
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Probability and Statistics
by
Milicevic3306
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15.8k
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17
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gate2016ec1
numericalanswers
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poissondistribution
randomvariable
0
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17
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 _________
asked
Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
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15.8k
points)

21
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gate2016ec1
numericalanswers
probabilityandstatistics
probability
0
votes
0
answers
18
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 _______.
asked
Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

10
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gate2015ec3
numericalanswers
probabilityandstatistics
probability
expectation
0
votes
0
answers
19
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 __________.
asked
Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

19
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gate2015ec3
numericalanswers
probabilityandstatistics
propability
randomvariable
variance
0
votes
0
answers
20
GATE ECE 2015 Set 3  Question: 52
A random binary wave $y(t)$ is given by $y(t) = \sum_{n = \infty}^{\infty}X_{n}\:p(tnT\phi)$ where $p(t)=u(t)u(tT),u(t)$ 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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Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

13
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gate2015ec3
numericalanswers
probabilityandstatistics
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uniformdistribution
0
votes
0
answers
21
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 _______.
asked
Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

10
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gate2015ec2
numericalanswers
probabilityandstatistics
probability
randomvariable
expectation
0
votes
0
answers
22
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 $pq+(1p)(1q)$ $pq$ $p(1q)$ $1pq$
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Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

11
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gate2015ec2
probabilityandstatistics
probability
randomvariable
0
votes
0
answers
23
GATE ECE 2015 Set 1  Question: 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? $P(A \cap B) = P(A)P(B)$ $P(A \mid B) = P(A)$ $P(A \cup B) = P(A) + P(B)$ $P(\overline{A} \cap \overline{B} )= P(\overline{A})P(\overline{B})$
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Mar 28, 2018
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Probability and Statistics
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Milicevic3306
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15.8k
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16
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gate2015ec1
probabilityandstatistics
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independentevents
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0
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24
GATE ECE 2015 Set 1  Question: 49
The input $X$ to the Binary Symmetric Channel (BSC) shown in the figure is $’1’$ with probability $0.8$. The crossover 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
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15.8k
points)

27
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gate2015ec1
numericalanswers
probabilityandstatistics
probability
0
votes
0
answers
25
GATE ECE 2015 Set 1  Question: 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 ... $0$ $1/12$ $1/9$ $1/6$
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Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
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15.8k
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24
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gate2015ec1
probabilityandstatistics
probability
probabilitydensityfunction
0
votes
0
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26
GATE ECE 2014 Set 4  Question: 27
Parcels from sender S to receiver R pass sequentially through twopost offices. Each postoffice 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)

23
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gate2014ec4
numericalanswers
probabilityandstatistics
probability
0
votes
0
answers
27
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 ___________
asked
Mar 26, 2018
in
Probability and Statistics
by
Milicevic3306
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15.8k
points)

9
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gate2014ec4
numericalanswers
probabilityandstatistics
probability
0
votes
0
answers
28
GATE ECE 2014 Set 3  Question: 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)

20
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gate2014ec3
probabilityandstatistics
probability
0
votes
0
answers
29
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 _______ .
asked
Mar 26, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)

10
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gate2014ec3
probabilityandstatistics
probability
expectation
numericalanswers
0
votes
0
answers
30
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 _________.
asked
Mar 26, 2018
in
Probability and Statistics
by
Milicevic3306
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15.8k
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28
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gate2014ec3
probabilityandstatistics
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0
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31
GATE ECE 2014 Set 3  Question: 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
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Milicevic3306
(
15.8k
points)

25
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gate2014ec3
probabilityandstatistics
probability
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0
votes
0
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32
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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Mar 26, 2018
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Probability and Statistics
by
Milicevic3306
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15.8k
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18
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gate2014ec2
probabilityandstatistics
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0
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0
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33
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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Mar 26, 2018
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Probability and Statistics
by
Milicevic3306
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15.8k
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25
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gate2014ec2
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0
votes
0
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34
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 ________.
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Mar 26, 2018
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Probability and Statistics
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Milicevic3306
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15.8k
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26
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gate2014ec1
numericalanswers
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0
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0
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35
GATE ECE 2014 Set 1  Question: 49
Let $X$ be a realvalued 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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Mar 26, 2018
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gate2014ec1
probabilityandstatistics
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0
votes
0
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36
GATE ECE 2014 Set 1  Question: 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 autocorrelation $E[X(t_{1})X(t_{2})]$ is $\cos(2\pi(t_{1} + t_{2}))$ $\sin(2\pi(t_{1}  t_{2}))$ $\sin(2\pi(t_{1} + t_{2}))$ $\cos(2\pi(t_{1}  t_{2}))$
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Mar 26, 2018
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Probability and Statistics
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Milicevic3306
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15.8k
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17
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gate2014ec1
probabilityandstatistics
statistics
uniformdistribution
correlationandregressionanalysis
0
votes
0
answers
37
GATE ECE 2013  Question: 38
Consider two identically distributed zeromean 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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Mar 26, 2018
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Probability and Statistics
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Milicevic3306
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15.8k
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34
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gate2013ec
probabilityandstatistics
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randomvariable
0
votes
0
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38
GATE ECE 2013  Question: 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 $4/9$ $1/2$ $2/3$ $5/9$
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Mar 26, 2018
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Probability and Statistics
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15
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gate2013ec
probabilityandstatistics
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0
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0
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39
GATE ECE 2012  Question: 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 $\frac{7}{80}$ $\frac{63}{80}$ $\frac{9}{10}$ $\frac{1}{10}$
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Mar 25, 2018
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Probability and Statistics
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Milicevic3306
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44
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gate2012ec
probabilityandstatistics
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40
GATE ECE 2012  Question: 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 $\frac{3}{4}$ $\frac{9}{16}$ $\frac{1}{4}$ $\frac{2}{3}$
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Mar 25, 2018
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Probability and Statistics
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15.8k
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