Previous Year Questions- Probability and Statistics - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

Q1: Let X be a discrete random variable that is uniformly distributed over the set {−10, −9, …, 0,…, 9, 10}. Which of the following random variables is/are uniformly distributed?       (2024)
(a) X²
(b) X³
(c) (X - 5)²
(d) (X + 10)²
Ans: (b, d)
Sol: So, X² is not uniformly distributed.
(X−5)² = {225, 199,….4, 1, 0, 1, 4, 9, 16, 25}
Again probability of choosing any number is not equal so (X−5)² is also not uniformly - distributed.
(X+10)² = {0, 1, 4, 9, 16,… 81, 100, 121,…. 400}  
P (Choosing any number) = 1/21 = so is it uniformly-distributed
Now,  X³ = {−1000, − 729, −512,…,−8, −1, 0, 1, 8, … 729, 1000}
Again P (Choosing any number) = 1/21 =  Constant
So, it is also uniformly-distributed.

Q2: The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is ___ (Round off to 3 decimal places).      (2023)
(a) 0.125
(b) 0.188
(c) 0.254
(d) 0.564
Ans: (b)
Sol: Let probability of head = P
Let probability of Tail  = q = P - 1
∴ Expected No. of trail
Given : Trial = 4
Now, probability of head for second trail.
= qP
Q3: Let the probability density function of a random variable x be given as
f(x) = ae^−2∣x∣
The value of ′a′ is _________         (2022)
(a) 0.5
(b) 1
(c) 1.5
(d) 2
Ans: (b)
Sol: Therefore,
a = 1

Q4: Suppose the probability that a coin toss shows "head" is p, where 0 < p < 1. The coin is tossed repeatedly until the first "head" appears. The expected number of tosses required is      (2021)
(a) $𝑝/(1-𝑝)$p/(1−p)
(b) $(1-𝑝)/𝑝$(1−p)/p
(c) 1/p
(d) 1/p²
Ans: (c)
Sol: 
Q5: The probability of a resistor being defective is 0.02. There are 50 such resistors in a circuit. The probability of two or more defective resistors in the circuit (round off to two decimal places)is ________      (2019)
(a) 0.1
(b) 0.26
(c) 0.65
(d) 0.85
Ans: (b)
Sol: 
Q6: The mean-square of a zero-mean random process is kT/C, where k is Boltzmann's constant, T is the absolute temperature, and C is a capacitance. The standard deviation of the random process is        (2019)
(a) kT/C
(b)
(c) C/KT
(d)
Ans: (b)
Sol: Given that,
Standard deviation =

Q7: A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var {Y}, where var {.} denotes the variance, equals      (SET-2 (2017))
(a) 7/8
(b) 49/64
(c) 7/64
(d) 105/64
Ans: (c)
Sol: 
Q8: Assume that in a traffic junction, the cycle of the traffic signal lights is 2 minutes of green (vehicle does not stop) and 3 minutes of red (vehicle stops). Consider that the arrival time of vehicles at the junction is uniformly distributed over 5 minute cycle. The expected waiting time (in minutes) for the vehicle at the junction is ________.       (SET-2 (2017))
(a) 0.4
(b) 0.9
(c) 1.5
(d) 2.6
Ans: (b)
Sol: t be arrival time of vehicles of the junction is uniformaly distributed in [0, 5].
Let y be the waiting time of the junction.

Q9: An urn contains 5 red balls and 5 black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball in the second draw is       (SET-2 (2017))
(a) 1/2
(b) 4/9
(c) 5/9
(d) 6/9
Ans: (a)
Sol: 
Q10: Let the probability density function of a random variable, X, be given as: where u(x) is the unit step function.
Then the value of 'a' and  Prob{X ≤ 0}, respectively, are      (SET-2 (2016))
(a) 2, 1/2
(b) 4, 1/2
(c) 2, 1/4
(d) 4, 1/4
Ans: (a)
Sol: 
Q11: Candidates were asked to come to an interview with 3 pens each. Black, blue, green and red were the permitted pen colours that the candidate could bring. The probability that a candidate comes with all 3 pens having the same colour is _____.       (SET-1(2016))
(a) 0.1
(b) 0.2
(c) 0.4
(d) 0.8
Ans: (b)  

Q12: Two players, A and B, alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player A starts the game, the probability that A wins the game is      (SET-1(2015))
(a) 5/11
(b) 1/2
(c) 7/13
(d) 6/11
Ans: (d)
Sol: P(A wins) = P( 6 in first throw by A) + P(A not 6, B not 6,) + ...

Q13: A random variable X has probability density function f(x) as given below:
If the expected value E[X] = 2/3, then $𝑃𝑟[𝑋<0.5]$Pr[X < 0.5] is ______.       (SET-1(2015))
(a) 0.25
(b) 0.5
(c) 0.75
(d) 1
Ans: (a)
Sol: 
Q14: Lifetime of an electric bulb is a random variable with density f(x) = kx², where x is measured in years. If the minimum and maximum lifetimes of bulb are 1 and 2 years respectively, then the value of k is _____        (SET-3 (2014))
(a) 0.85
(b) 0.42
(c) 0.25
(d) 0.75
Ans: (b)
Sol: Life time of an electric bulb with density
If minimum and maximum lifetimes of bulb are 1 and 2 years respectively then

Q15: Let X be a random variable with probability density function The probability P(0.5 < X < 5) is        (SET-2 (2014))
(a) 0.15
(b) 0.40
(c) 0.75
(d) 0.85
Ans: (b)
Sol: Probability (0.5 < n < 5)