Probability and Statistics · Engineering Mathematics · GATE ECE
Marks 1
1
Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $X \geq Y$ is _______ .
GATE ECE 2024
2
The bar graph shown the frequency of the number of wickets taken in a match by a bowler in her career. For example, in 17 of her matches, the bowler has taken 5 wickets each. The median number of wickets taken by the bowler in a match is ____________ (rounded off to one decimal place).
GATE ECE 2022
3
Let
X1
, X2
, X3
and
X4
be independent normal random variables with zero mean and unit
variance. The probability that
X4
is the smallest among the four is _______.
GATE ECE 2018
4
$$500$$ students are taking one or more courses out of chemistry, physics and Mathematics. Registration records indicate course enrolment as follows: chemistry $$(329)$$, physics $$(186)$$, Mathematics $$(295)$$, chemistry and physics $$(83)$$, chemistry and Mathematics $$(217)$$, and physic and Mathematics $$(63)$$, How many students are taking all $$3$$ subjects?
GATE ECE 2017 Set 2
5
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) _________.
GATE ECE 2017 Set 1
6
The second moment of a Poisson-distributed random variables is $$2.$$ The mean of the random variable is _______.
GATE ECE 2016 Set 1
7
The variance of the random variable $$X$$ with probability density function $$\,f\left( x \right) = {1 \over 2}\left| x \right|{e^{ - \left| x \right|}}\,\,$$ is ___________.
GATE ECE 2015 Set 3
8
Ram and Ramesh appeared in an interview for two vacancies in the same department. The probability of Ram's selection is $$1/6$$ and that of Ramesh is $$1/8$$. What is the probability that only one of them will be selected?
GATE ECE 2015 Set 2
9
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 ________.
GATE ECE 2015 Set 2
10
Suppose $$A$$ & $$B$$ are two independent events with probabilities $$P\left( A \right) \ne 0$$ and $$P\left( B \right) \ne 0.$$ Let $$\overrightarrow A $$ & $$\overrightarrow B $$ be their complements. Which of the following statements is FALSE?
GATE ECE 2015 Set 1
11
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 _______.
GATE ECE 2014 Set 1
12
Let $$X$$ be a real - valued random variable with $$E\left[ X \right]$$ and $$E\left[ {{X^2}} \right]$$ denoting the mean values of $$X$$ and $${{X^2}}$$, respectively. The relation which always holds true is
GATE ECE 2014 Set 1
13
If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier of future calls, the probability distribution function of the total number of calls in a fixed time interval will be
GATE ECE 2014 Set 4
14
Let $$X$$ be a zero mean unit variance Gaussian random variable. $$E\left[ {\left| X \right|} \right]$$ is equal to ______
GATE ECE 2014 Set 4
15
An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is
GATE ECE 2014 Set 3
16
Two independent random variables $$X$$ and $$Y$$ are uniformly distributed in the interval $$\left[ { - 1,1} \right].$$ The probability that max $$\left[ {X,Y} \right]$$ is less than $$1/2$$ is
GATE ECE 2012
17
A fair dice is tossed two times. The probability that the $$2$$nd toss results in a value that is higher than the first toss is
GATE ECE 2011
18
A fair coin is tossed $$10$$ times. What is the probability that only the first two tosses will yield heads?
GATE ECE 2009
19
If $$E$$ denotes expectation, the variance of a random variable $$X$$ is given by
GATE ECE 2007
20
A fair dice is rolled twice. The probability that an odd number will follow an even number is
GATE ECE 2005
Marks 2
1
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 ___________.
GATE ECE 2017 Set 2
2
Two random variables $$X$$ and $$Y$$ are distributed according to
$$${f_{X,Y}}\left( {x,y} \right) = \left\{ {\matrix{
{\left( {x + y} \right),} & {0 \le x \le 1,} & {0 \le y \le 1} \cr
{0,} & {otherwise} & \, \cr
} } \right.$$$
The probability $$P\left( {X + Y \le 1} \right)$$ is ________.
GATE ECE 2016 Set 1
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 ________.
GATE ECE 2016 Set 3
4
A fair die with faces $$\left\{ {1,2,3,4,5,6} \right\}$$ is thrown repeatedly till $$'3'$$ is observed for the first time. Let $$X$$ denote the number of times the dice is thrown. The expected value of $$X$$ is _________.
GATE ECE 2015 Set 3
5
Let $$\,\,X \in \left\{ {0,1} \right\}\,\,$$ and $$\,\,Y \in \left\{ {0,1} \right\}\,\,$$ be two independent binary random variables. If $$\,\,P\left( {X\,\, = 0} \right) = p\,\,$$ and $$\,\,P\left( {Y\,\, = 0} \right) = q\,\,$$, then $$P\left( {X + Y \ge 1} \right)$$ is equal to
GATE ECE 2015 Set 2
6
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 _______.
GATE ECE 2015 Set 1
7
Let $$\,{X_1},\,\,{X_2}\,\,$$ and $$\,{X_3}\,$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,1} \right].$$ The probability $$p$$ {$${X_1}$$ is the largest} is __________.
GATE ECE 2014 Set 1
8
Parcels from sender $$S$$ to receiver $$R$$ pass sequentially through two post - offices. Each post - office has a probability $${1 \over 5}$$ 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 _________.
GATE ECE 2014 Set 4
9
Let $${X_1},{X_{2,}}$$ and $${X_{3,}}$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,1} \right].$$ The probability $$P\left\{ {{X_1} + {X_2} \le {X_3}} \right\}$$ is _______.
GATE ECE 2014 Set 3
10
A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is ________.
GATE ECE 2014 Set 3
11
Let $$X$$ be a random variable which is uniformly chosen from the set of positive odd numbers less than $$100.$$ The expectation, $$E\left[ X \right],$$ is __________.
GATE ECE 2014 Set 2
12
Let $$U$$ and $$V$$ be two independent zero mean Gaussian random variables of variances $${1 \over 4}$$ and $${1 \over 9}$$ respectively. The probability $$\,P\left( {3V \ge 2U} \right)\,\,$$ is
GATE ECE 2013
13
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 $$x$$
GATE ECE 2013
14
A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is
GATE ECE 2012
15
A fair coin is tossed independently four times. The probability of the event ''The number of times heads show up is more than the number of times tails show up'' is
GATE ECE 2010
16
Consider two independent random variables $$X$$ and $$Y$$ with identical distributions. The variables $$X$$ and $$Y$$ take values $$0, 1$$ and $$2$$ with probability $$1/2,$$ $$1/4$$ and $$1/4$$ respectively. What is the conditional probability $$P(X+Y=2/X-Y=0)?$$
GATE ECE 2009
17
A discrete random variable $$X$$ takes value from $$1$$ to $$5$$ with probabilities as shown in the table. A student calculates the mean of $$X$$ as $$3.5$$ and her teacher calculates the variance to $$X$$ as $$1.5.$$ Which of the following statements is true?
GATE ECE 2009
18
An examination consists of two papers, paper $$1$$ and paper $$2.$$ The probability of failing in paper $$1$$ is $$0.3$$ and that in paper $$2$$ is $$0.2.$$ Given that a student has failed in paper $$2,$$ the probability of failing in paper $$1$$ is $$0.6.$$ The probability of a student failing in both the papers is
GATE ECE 2007