SYLLABUS
Counting (permutation and combinations), probability axioms, Sample space, events, independent events, mutually exclusive events, marginal, conditional and joint probability, Bayes Theorem, conditional expectation and variance, mean, median, mode and standard deviation, correlation, and covariance, random variables, discrete random variables and probability mass functions, uniform, Bernoulli, binomial distribution, Continuous random variables and probability distribution function, uniform, exponential, Poisson, normal, standard normal, t-distribution, chi-squared distributions, cumulative distribution function, Conditional PDF, Central limit theorem, confidence interval, z-test, t-test, chi-squared test
Q1 – Consider the following statements:
(i) The mean and variance of a Poisson random variable are equal.
(ii) For a standard normal random variable, the mean is zero and the variance is one.
Which ONE of the following options is correct?
(A) Both (i) and (ii) are true
(B) (i) is true and (ii) is false
(C) (ii) is true and (i) is false
(D) Both (i) and (ii) are false
(GATE DS&AI 2024)
Ans – (A)
Explanation – Events that occurred in a fixed amount of time or space are counted by Poisson distribution.
For example, around 5 cars cross in a signal within an average of 1 minute.
Thus, the average number of cars is 5 and so must be variance.
In other words, the statement (i) is true.
The normal distribution also known as the “bell curve” shape, is probably one of the most well know distributions in statistics.
Standard normal means that the curve is centred at 0. (Mean = 0)
No stretching, no compression. (Variance = 1)
It means statement (ii) is also correct.
Q2 – Three fair coins are tossed independently. T is the event that two or more tosses result in heads. S is the event that two or more tosses result in tails.
What is the probability of the event 𝑇∩𝑆 ?
(A) 0
(B) 0.5
(C) 0.25
(D) 1
(GATE DS&AI 2024)
Ans – (A)
Explanation – 3 tosses are there and T is the event that two or more tosses result in heads and S is the event that two or more tosses result in tails. If we look at the event 𝑇∩𝑆, it means four tosses should be there. 3 tosses can’t have both 2 heads and 2 tails.
So, it means probability is 0 (either T or S will make it zero).
Q3 – Let the minimum, maximum, mean and standard deviation values for the attribute income of data scientists be ₹46000, ₹170000, ₹96000, and ₹21000, respectively. The z-score normalized income value of ₹106000 is closest to which ONE of the following options?
(A) 0.217
(B) 0.476
(C) 0.623
(D) 2.304
(GATE DS&AI 2024)
Ans – (B)
Explanation – The z-score normalized income value = (106000 – mean)/standard deviation
= (106000 – 96000)/21000
= 0.476 is the answer (option B).
Q4 – The sample average of 50 data points is 40. The updated sample average after including a new data point taking the value of 142 is ______.
(GATE DS&AI 2024)
Ans – (42)
Explanation – Sample average of 50 data points = 40
Let’s say there are 50 data points each of 40 values.
51th data points is 142.
Average = (50*40 + 142)/51 = 42
Q5 – A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown independently.
What is the expected number of times the die is thrown until two consecutive throws of even numbers are seen?
(A) 2
(B) 4
(C) 6
(D) 8
(GATE DS&AI 2024)
Ans – (C)
Explanation –
Q6 – Let 𝑋 be a random variable uniformly distributed in the interval [1, 3] and 𝑌 be a random variable uniformly distributed in the interval [2, 4]. If X and Y are independent of each other, the probability P(𝑋≥𝑌) is ______ (rounded off to three decimal places).
(GATE DS&AI 2024)
Ans – (0.125)
Explanation –
Plot X on the x-axis from 1 to 3.
Plot Y on the y-axis from 2 to 4.
This will form a rectangle.
Now we generate a line x = y and mark the region where x ≥ y. This will form a right-angled triangle in the rectangle.
Area of triangle = ½*1*1 = 0.5
Then probability P(𝑋≥𝑌) is 0.5/Total Area = 0.5/4 = 0.125
Q7 – Let 𝑋 be a random variable exponentially distributed with parameter 𝜆 > 0. The probability density function of X is given by:
If 5𝐸(𝑋) = 𝑉𝑎𝑟(𝑋), where 𝐸(𝑋) and 𝑉𝑎𝑟(𝑋) indicate the expectation and variance of 𝑋, respectively, the value of 𝜆 is ______ (rounded off to one decimal place).
(GATE DS&AI 2024)
Ans – (0.2)
Explanation – E(X) = 1/ 𝜆
Var(x) = 1/ 𝜆2
If 5.E(X) = Var(X),
Then 5.(1/𝜆) = (1/ 𝜆2)
=> 𝜆 = 0.2
Q8 – Consider two events T and S. Let 𝑇’ denote the complement of the event T. The probability associated with different events are given as follows: 𝑃(𝑇’)=0.6, 𝑃(𝑆|𝑇)=0.3, 𝑃(𝑆|𝑇‘)=0.6
Then, 𝑃(𝑇|𝑆) is ______ (rounded off to two decimal places).
(GATE DS&AI 2024)
Ans – (0.25)
Explanation – P(T|S) = (P(S|T)*P(T)) / P(S)
P(T|S) = 0.3*(1 – P(T’)) / [ P(S∣T)*P(T) + P(S∣T′)*P(T′) ]
P(T|S) = (0.3*0.4) / [ 0.3*0.4 + 0.6*0.6 ]
P(T|S) = 0.12 / 0.48 = 0.25
Q9 – Consider a joint probability density function of two random variables X and Y
Then, 𝐸[𝑌|𝑋=1.5] is ______.
(GATE DS&AI 2024)
Ans – (1)
Q10 – Two fair coins are tossed independently. X is a random variable that takes a value of 1 if both tosses are heads and 0 otherwise. Y is a random variable that takes a value of 1 if at least one of the tosses is heads and 0 otherwise.
The value of the covariance of X and Y is ______ (rounded off to three decimal places).
(GATE DS&AI 2024)
Ans – (0.062 to 0.063)
Explanation –
Event | Probability | X | Y |
HH | 1/4 | 1 | 1 |
HT | 1/4 | 0 | 1 |
TH | 1/4 | 0 | 1 |
TT | 1/4 | 0 | 0 |
Expected Value of X = ¼
Expected Value of Y = ¾
Expected Value of XY (when X = 1 & Y = 1) = ¼
Covariance of X and Y = Expected Value of XY – (Expected Value of X)(Expected Value of Y)
Covariance of X and Y = 1/4 – (1/4*3/4) = 1/16 = 0.063.