GATE EC ||COMMUNICATION SYSTEM||RANDOM VARIABLE|| PYQs (2000-2025)

Consider two identically 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

Let X 1 , X 2 , X 3  and X 4  be independent normal random variables with zero mean and unit variance. The probability that X 4  is the smallest among the four is ___________.(rounded off to two decimal places)

(PYQ || 2008 NAT|| 1 MARK)

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 of die is thrown. The expected value of X is________.

(PYQ || 2015 NAT|| 2 MARK)

The variance of the random variable X with probability density function f(x) =1/2|x|e^-|x| is ________.

(PYQ || 2015 NAT|| 2 MARK)

(PYQ || 2015 MCQ|| 2 MARK)

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 ______.(rounded off to one decimal place)

(PYQ || 2015 NAT|| 1 MARK)

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 or future calls, the probability distribution function of the total number of calls in a fixed time interval will be

(PYQ || 2014 MCQ|| 1 MARK)

Let, X be a zero mean unit variance Gaussian random variable. E[|X|] is equal to ___________.(rounded off to two decimal place)

( PYQ || 2014 NAT|| 2 MARK)

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 variable Y1 and Y2 as shown in the figure.

The value of H(Y ₁ ) + H(Y ₂ ) in bits is _____.

(PYQ || 2014 NAT|| 2 MARK)

Let X ₁ , X ₂ and X ₃ be independent and identically distributed random variable with the uniform distribution on [0, 1]. The probability P(X ₁ + X ₂ < X ₃ ) is

_________. (rounded off to two decimal place)

(PYQ || 2014 NAT|| 2 MARK)