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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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Official PYQ2017Engineering MathematicsProbability and StatisticsNAT · 1 mark
Consider the random process X(t)=U+Vt,X(t)=U+Vt, Where UU is a zero-mean Gaussian random variable and V is a random variable uniformly distributed between 00 and 22. Assume that UU and VV are statistically independent. The mean value of the random process at t=2t = 2 is ________
Official PYQ2017Engineering MathematicsProbability and StatisticsNAT · 2 marks
Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is 40%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 __________
Official PYQ2016Engineering MathematicsProbability and StatisticsNAT · 1 mark
The second moment of a Poisson-distributed random variable is 22. The mean of the random variable is _____
Official PYQ2016Engineering MathematicsProbability and StatisticsNAT · 2 marks
Consider a discrete memoryless source with alphabet S={s0,s1,s2,s3,s4,… }S = \{s_0,s_1,s_2,s_3,s_4, \dots \} and respective probabilities of occurence P={12,14,18,116,132,…}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 _________
Official PYQ2016Engineering MathematicsProbability and StatisticsMCQ · 2 marks
An analog pulse s(t)s(t) is transmitted over an additive white Gaussian noise (AWGN) channel. The received signal is r(t)=s(t)+n(t)r(t) = s(t) + n(t), where n(t)n(t) is additive white Gaussian noise with power spectral density N02\frac{N_0}{2}. The received signal is passed through a filter with impulse response h(t)h(t). Let EsE_s and EhE_h denote the energies …
Official PYQ2016Engineering MathematicsProbability and StatisticsNAT · 1 mark
A discrete memoryless source has an alphabet {a1,a2,a3,a4}\left \{ a_{1},a_{2}, a_{3},a_{4}\right \} with corresponding probabilities {12,14,18,18}.\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 error-free reconstruction is _________
Official PYQ2016Engineering MathematicsProbability and StatisticsNAT · 2 marks
Two random variables XX and YY are distributed according to fX,Y(x,y)={(x+y),0≤x≤1,0≤y≤10,otherwise.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≤1)P(X+Y\leq 1) is ________
Official PYQ2016Engineering MathematicsProbability and StatisticsNAT · 1 mark
The probability of getting a “head” in a single toss of a biased coin is 0.30.3. 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 _______
Official PYQ2016Engineering MathematicsProbability and StatisticsNAT · 2 marks
A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of 4.0 kHz4.0\:kHz and two-sided noise power spectral density η2=2.5×10−5Watt per Hz{\large\frac{\eta}{2}}=2.5\times10^{-5}Watt\:per\:Hz. If information at the rate of 52 kbps52\:kbps is to be transmitted over this channel with arbitrarily small bit error rate, then the minimum bit-energy EbE_b …
Official PYQ2016Engineering MathematicsProbability and StatisticsNAT · 2 marks
The bit error probability of a memoryless binary symmetric channel is 10−510^{-5}. If 10510^5 bits are sent over this channel, then the probability that not more than one bit will be in error is _______
Official PYQ2015Engineering MathematicsProbability and StatisticsMCQ · 1 mark
Suppose AA and BB are two independent events with probabilities P(A)≠0P(A) \neq 0 and P(B)≠0P(B) \neq 0. Let A‾\overline{A} and B‾\overline{B} be their complements. Which one of the following statements is FALSE?
Official PYQ2015Engineering MathematicsProbability and StatisticsNAT · 2 marks
The input XX to the Binary Symmetric Channel (BSC) shown in the figure is ’1’’1’ with probability 0.80.8. The cross-over probability is 1/71/7. If the received bit Y=0Y=0, the conditional probability that ’1’’1’ was transmitted is ____________
Official PYQ2015Engineering MathematicsProbability and StatisticsMCQ · 2 marks
A source emits bit 00 with probability 13\frac{1}{3} and bit 11 with probability 23\frac{2}{3}. The emitted bits are communicated to the receiver. The receiver decides for either 00 or 11 based on the received value RR. It is given that the conditional density functions of RR are as …
Official PYQ2015Engineering MathematicsProbability and StatisticsNAT · 2 marks
Let the random variable XX represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of XX is _______.
Official PYQ2015Engineering MathematicsProbability and StatisticsMCQ · 2 marks
A zero mean white Gaussian noise having power spectral density N02\dfrac{N_{0}}{2} is passed through an LTI filter whose impulse response h(t)h(t) is shown in the figure. The variance of the filtered noise at t=4t = 4 is
Official PYQ2015Engineering MathematicsProbability and StatisticsMCQ · 2 marks
{Xn}n=−∞n=∞\begin{Bmatrix} X_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty} is an independent and identically distributed (i.i.d.) random process with ܺXnX_{n} equally likely to be +1+1 or −1.  {Yn}n=−∞n=∞−1.\:\:\begin{Bmatrix} Y_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty} is another random process obtained as ܻYn=Xn+0.5Xn−1.Y_{n}=X_{n} + 0.5X_{n-1}. The autocorrelation function of …
Official PYQ2015Engineering MathematicsProbability and StatisticsMCQ · 2 marks
Let X∈{0,1}X\in \{0,1\} and Y∈{0,1}Y\in \{0,1\} be two independent binary random variables. If P(X=0)=pP(X=0)=p and P(Y=0)=q,P(Y=0)=q, then P(X+Y≥1)P(X+Y\geq 1) is equal to
Official PYQ2015Engineering MathematicsProbability and StatisticsNAT · 2 marks
A fair die with faces {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\} is thrown repeatedly till ’3’’3’ is observed for the first time. Let XX denote the number of times the die is thrown. The expected value of XX is _______.
Official PYQ2015Engineering MathematicsProbability and StatisticsNAT · 2 marks
The variance of the random variable XX with probability density function f(x)=12∣x∣e−∣x∣f(x)=\dfrac{1}{2}\mid x \mid e^{- \mid x \mid} is __________.
Official PYQ2015Engineering MathematicsProbability and StatisticsNAT · 2 marks
A random binary wave y(t)y(t) is given by y(t)=∑n=−∞∞Xn p(t−nT−ϕ)y(t) = \sum_{n = -\infty}^{\infty}X_{n}\:p(t-nT-\phi) where p(t)=u(t)−u(t−T),u(t)p(t)=u(t)-u(t-T),u(t) is the unit step function and ϕ\phi is an independent random variable with uniform distribution in [0,T].[0,T].The sequence {Xn}\{X_{n}\} consist of independent and identically distributed binary valued random variables with …