This set of Probability and Statistics Multiple Choice Questions & Answers (MCQs) focuses on “Mean and Variance of Distribution”.

1. The expectation of a random variable X (E(X)) can be written as _________

a) \(\frac{d}{dt} [M_X (t)](t=0) \)

b) \(\frac{d}{dx} [M_X (t)](t=0) \)

c) \(\frac{d^2}{dt^2} [M_X (t)](t=0) \)

d) \(\frac{d^2}{dx^2} [M_X (t)](t=0) \)

View Answer

Explanation: Expectation of a random variable X can be written as the first differentiation of Moment generating function, which can be written as \(\frac{d}{dt} [M_X (t)](t=0). \)

2. If the probability of hitting the target is 0.4, find mean and variance.

a) 0.4, 0.24

b) 0.6, 0.24

c) 0.4, 0.16

d) 0.6, 0.16

View Answer

Explanation: p = 0.4

q = 1-p

= 1-0.4 = 0.6

Therefore, mean = p = 0.4 and

Variance = pq = (0.4) (0.6) = 0.24.

3. If the probability that a bomb dropped from a place will strike the target is 60% and if 10 bombs are dropped, find mean and variance?

a) 0.6, 0.24

b) 6, 2.4

c) 0.4, 0.16

d) 4, 1.6

View Answer

Explanation: Here, p = 60% = 0.6 and q = 1-p = 40% = 0.4 and n = 10

Therefore, mean = np = 6

Variance = npq = (10)(0.6)(0.4)

= 2.4.

4. If P(1) = P(3) in Poisson’s distribution, what is the mean?

a) \(\sqrt{2} \)

b) \(\sqrt{3} \)

c) \(\sqrt{6} \)

d) \(\sqrt{7} \)

View Answer

Explanation: \(P(x) = \frac{(e^{-λ} λ^x)}{x!} \)

Therefore, \(P(3) = \frac{(e^{-λ} λ^3)}{3!} \)

and \(P(1) = \frac{(e^{-λ} λ^1)}{1!} \)

P(1) = P(2)

\(λ=\frac{λ^3}{6} \)

Therefore, \(λ=\sqrt{6}. \)

5. What is the mean and variance for standard normal distribution?

a) Mean is 0 and variance is 1

b) Mean is 1 and variance is 0

c) Mean is 0 and variance is ∞

d) Mean is ∞ and variance is 0

View Answer

Explanation: The mean and variance for the standard normal distribution is 0 and 1 respectively.

6. Find λ in Poisson’s distribution if the probabilities of getting a head in biased coin toss as \(\frac{3}{4} \) and 6 coins are tossed.

a) 3.5

b) 4.5

c) 5.5

d) 6.6

View Answer

Explanation: p =

^{3}⁄

_{4}

λ = np = (6)

^{3}⁄

_{4}= 4.5.

7. If P(6) = λP(1) in Poisson’s distribution, what is the mean?(Approximate value)

a) 4

b) 6

c) 5

d) 7

View Answer

Explanation: \(\frac{e^{-λ} λ^6}{6!}= λ \frac{e^{-λ} λ^1}{1!} \)

λ

^{4}= 6! = 720

Therefore λ = 5.18 = 5.

8. Find f(2) in normal distribution if mean is 0 and variance is 1.

a) 0.1468

b) 0.1568

c) 0.1668

d) 0.1768

View Answer

Explanation: Given mean = 0

Variance = 1

\(f(2) = \frac{1}{(\sqrt{2π})} e^{\frac{-1}{2} \frac{2}{1}}= 0.1468. \)

9. Find the mean of tossing 8 coins.

a) 2

b) 4

c) 8

d) 1

View Answer

Explanation: p =

^{1}⁄

_{2}

n = 8

q =

^{1}⁄

_{2}

Therefore, mean = np = 8 *

^{1}⁄

_{2}= 4.

10. Mean and variance of Poisson’s distribution is the same.

a) True

b) False

View Answer

Explanation: The mean and variance of Poisson’s distribution are the same which is equal to λ.

**Sanfoundry Global Education & Learning Series – Probability and Statistics.**

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