GATE DA Probability and Statistics Quiz

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Question 1

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? 

  • Both (i) and (ii) are true

  • (i) is true and (ii) is false

  • (ii) is true and (i) is false

  • Both (i) and (ii) are false

Question 2

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 that (T ∧ S) ?

  • 0

  • 0.5

  • 0.25

  • 1

Question 3

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?
 

  • 2

  • 4

  • 6

  • 8

Question 4

Let 𝑋 be a random variable uniformly distributed in the interval [1, 3] and π‘Œ be a random variable uniformly distributed in the interval [2, 4]. If X and Y are independent of each other, the probability P(𝑋 β‰₯ π‘Œ) is ______ (rounded off to three decimal places)

  • 0.167

  • 0.134

  • 0.125

  • 0.215

Question 5

Let 𝑋 be a random variable exponentially distributed with parameter πœ† > 0. The probability density function of X is given by: 

𝑓𝑋 (π‘₯) = πœ† 𝑒 βˆ’πœ†π‘₯ , π‘₯ β‰₯ 0 
𝑓𝑋 (π‘₯) =0, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’

If 5𝐸(𝑋) = π‘‰π‘Žπ‘Ÿ(𝑋), where 𝐸(𝑋) and π‘‰π‘Žπ‘Ÿ(𝑋) indicate the expectation and variance of 𝑋, respectively, the value of πœ† is ______ (rounded off to one decimal place)

  • 0.2

  • 0.3

  • 0.1

  • 0.4

Question 6

Consider two events T and S. Let 𝑇̅denote the complement of the event T . The probability associated with different events are given as follows: 𝑃(~𝑇̅) = 0.6 , 𝑃(𝑆|𝑇) = 0.3, 𝑃(𝑆|~𝑇̅) = 0.6 Then, 𝑃(𝑇|𝑆) is ______ (rounded off to two decimal places).

  • 0.22

  • 0.25

  • 0.50

  • 0.28

Question 7

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)

  • 0.062

  • 0.072

  • 0.056

  • 0.076

Question 8

The probability of a boy or a girl being born is 1/2. For a family having only three children, what is the probability of having two girls and one boy?

  • 3/8

  • 1/8

  • 1/4

  • 1/2

Question 9

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?

  • Both (i) and (ii) are true

  • (i) is true and (ii) is false

  • (ii) is true and (i) is false

  • Both (i) and (ii) are false

Question 10

Consider five random variables π‘ˆ, 𝑉, π‘Š, 𝑋, and π‘Œ whose joint distribution satisfies:

𝑃(π‘ˆ, 𝑉, π‘Š, 𝑋, π‘Œ) = 𝑃(π‘ˆ)𝑃(𝑉)𝑃(π‘Š|π‘ˆ, 𝑉)𝑃(𝑋|π‘Š)𝑃(π‘Œ|π‘Š)

Which ONE of the following statements is FALSE?

  • π‘Œ is conditionally independent of 𝑉 given π‘Š

  • 𝑋 is conditionally independent of π‘ˆ given π‘Š

  • π‘ˆ and 𝑉 are conditionally independent given π‘Š

  • π‘Œ and 𝑋 are conditionally independent given W

There are 23 questions to complete.

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