GATE EC Probability and Statistics questions

Practise Probability and Statistics questions for GATE EC. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

Define the sample space and events first. Distinguish independent events from conditional probabilities, and check that the final probability lies between zero and one.

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SubjectTopicYearType

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 ________.

X_{1},X_{2},

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}\}

X

be a real-valued random variable with

E[X]

E[X^{2}]

denoting the mean values of

X

X^{2},

respectively. The relation which always holds true is

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})]

X

be a random variable which is uniformly chosen from the set of positive odd numbers less than

100

. The expectation,

E[X]

The capacity of a band-limited additive white Gaussian noise (AWGN) channel is given by

C = W \log_{2}\left ( 1+\frac{p} {\sigma ^{2}w} \right )

bits per second (bps), where

W

is the channel bandwidth,

P

is the average power received and

\sigma ^{2}

is the one-sided power spectral density of the AWGN. For a fixed …

1

– bit quantizer is a random variable

X

f_{X}( x )= 2e^{-2x}

x\geq 0

f_{X} (x )= 0

x< 0

. For outputs to be of equal probability, the quantizer threshold should be ______.

An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is _______ .

X_{1},X_{2},

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 \}

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}

Y_{2}

as shown in the figure. The value of

H( Y_{1} )+H( Y_{2} )

X

be a zero mean unit variance Gaussian random variable.

E[ \mid X \mid ]

is equal to __________

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

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 _________

Consider a communication scheme where the binary valued signal

X

P\{X=+1\}=0.75

P\{X=-1 \}=0.25

. The received signal

Y=X+Z

Z

is a Gaussian random variable with zero mean and variance

\sigma ^2

. The received signal

Y

is fed to the threshold detector. The output of the threshold detector

\hat{X}

Z

-channel given in the figure. The input is

0

1

with equal probability. If the output is

0

, the probability that the input is also

0

equals ___________

Consider a discrete-time channel

Y=X +Z

, where the additive noise

Z

is signal-dependent. In particular, given the transmitted symbol

X \in \{-a , +a\}

at any instant, the noise sample

Z

is chosen independently from a Gaussian distribution with mean

\beta X

and unit variance. Assume a threshold detector with zero threshold at the receiver. …

U

V

be two independent zero mean Gaussian random variables of variances

\dfrac{1}{4}

\dfrac{1}{9}

respectively. The probability

P(3V\geq 2U)

Consider two identically distributed zero-mean random variables

U

V.

Let the cumulative distribution functions of

U

2V

F(x)

G(x)

respectively. Then, for all values of

x

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