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Recent questions and answers in Probability and Statistics
+1
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1
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1
GATE201412
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)
gate2014ec1
numericalanswers
probabilityandstatistics
engineeringmathematics
0
votes
0
answers
2
GATE2020EC: 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 __________.
asked
Feb 13
in
Probability and Statistics
by
jothee
(
1.4k
points)
gate2020ec
numericalanswers
probabilityandstatistics
0
votes
1
answer
3
GATE201236
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)
gate2012ec
probability
engineeringmathematics
0
votes
0
answers
4
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 _________
asked
Nov 21, 2019
in
Probability and Statistics
by
KUSHAGRA गुप्ता
(
260
points)
gate2016ec
probability
0
votes
0
answers
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)
gate2009ec
probability
0
votes
0
answers
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)
gate2019ec
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
7
GATE2016351
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)
gate2016ec3
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
8
GATE2016221
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)
gate2016ec2
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
9
GATE2016228
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)
gate2016ec2
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
10
GATE201612
The second moment of a Poissondistributed random variable is $2$. The mean of the random variable is _____
asked
Mar 28, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)
gate2016ec1
numericalanswers
probabilityandstatistics
engineeringmathematics
0
votes
0
answers
11
GATE2016148
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)
gate2016ec1
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
12
GATE2015350
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)
gate2015ec3
numericalanswers
propability
engineeringmathematics
0
votes
0
answers
13
GATE201513
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)
gate2015ec1
probability
engineeringmathematics
0
votes
0
answers
14
GATE2015149
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
(
15.8k
points)
gate2015ec1
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
15
GATE2015152
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)
gate2015ec1
probability
engineeringmathematics
0
votes
0
answers
16
GATE201441
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)
gate2014ec4
probabilityandstatistics
engineeringmathematics
0
votes
0
answers
17
GATE2014427
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)
gate2014ec4
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
18
GATE2014450
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)
gate2014ec4
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
19
GATE201434
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)
gate2014ec3
probability
engineeringmathematics
0
votes
0
answers
20
GATE2014329
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)
gate2014ec3
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
21
GATE2014352
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)
gate2014ec3
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
22
GATE2014249
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)
gate2014ec2
numericalanswers
probabilityandstatistics
engineeringmathematics
0
votes
0
answers
23
GATE201415
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)
gate2014ec1
numericalanswers
probabilityandstatistics
engineeringmathematics
0
votes
0
answers
24
GATE2014123
The capacity of a Binary Symmetric Channel $\text{(BSC)}$ with crossover probability $0.5$ is ________.
asked
Mar 26, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)
gate2014ec1
numericalanswers
probabilityandstatistics
engineeringmathematics
0
votes
0
answers
25
GATE2014149
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 $(E[X])^{2}>E[X^{2}]$ $E[X^{2}]\geq (E[X])^{2}$ $E[X^{2}] = (E[X])^{2}$ $E[X^{2}] > (E[X])^{2}$
asked
Mar 26, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)
gate2014ec1
probabilityandstatistics
engineeringmathematics
0
votes
0
answers
26
GATE2014150
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 $\cos(2\pi(t_{1} + t_{2}))$ $\sin(2\pi(t_{1}  t_{2}))$ $\sin(2\pi(t_{1} + t_{2}))$ $\cos(2\pi(t_{1}  t_{2}))$
asked
Mar 26, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)
gate2014ec1
probabilityandstatistics
correlationandregressionanalysis
0
votes
0
answers
27
GATE201338
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 $x$ $F(x)  G(x) \leq 0$ $F(x)  G(x) \geq 0$ $(F(x)  G(x)) \cdot x\leq 0$ $(F(x)  G(x)) \cdot x\geq 0$
asked
Mar 26, 2018
in
Probability and Statistics
by
Milicevic3306
(
15.8k
points)
gate2013ec
probability
probabilityandstatistics
engineeringmathematics
0
votes
0
answers
28
GATE201326
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)
gate2013ec
probabilityandstatistics
probability
engineeringmathematics
0
votes
0
answers
29
GATE201238
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)
gate2012ec
probability
engineeringmathematics
0
votes
0
answers
30
GATE201224
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)
gate2012ec
probability
engineeringmathematics
0
votes
0
answers
31
GATE201215
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)
gate2012ec
probability
engineeringmathematics
0
votes
0
answers
32
GATE201840
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)
gate2018ec
numericalanswers
propability
engineeringmathematics
0
votes
0
answers
33
GATE201823
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)
gate2018ec
numericalanswers
probability
engineeringmathematics
+1
vote
0
answers
34
GATE2017 EC2: 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)
gate2017ec2
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
35
GATE2017 EC2: 22
Consider the random process $X(t)=U+Vt,$ Where $U$ is a zeromean 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)
gate2017ec2
numericalanswers
probability
engineeringmathematics
0
votes
0
answers
36
GATE2017 EC1: 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)
gate2017ec1
probability
numericalanswers
engineeringmathematics
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