Probability Questions and Answers – Mean and Variance of Distribution – Set 2

This set of Probability Multiple Choice Questions & Answers (MCQs) focuses on “Mean and Variance of Distribution – Set 2”.

1. The mean m of the distribution of X is defined by m=E(X)=∑xrfr, when ___________
a) X is discrete
b) X is continuous
c) X follows Normal distribution
d) X follows Poisson distribution
View Answer

Answer: a
Explanation: The mean m of a random variable X can only be specified in the summation form (i.e ∑xrfr) if and only if the random variable takes discrete values. If the random variable is continuous, then we need to calculate the integral over the interval area.

2. For a random variable X, Var(aX+b)=a2Var(x), is true or false?
a) True
b) False
View Answer

Answer: a
Explanation: my=E(y) = E(aX+b)=aE(X)+b=amx+b,
Var(aX+b)=Var(y)=E{(Y-my)2} = E{(aX+b-amx-b2)} = a2E{(x-mx)2} = a2Var(x), since Var(constant)=0.

3. The expectation E(x) of a continuous random variable X, can be expressed as ________ if F(x) is the distribution function of X.
a) E(X) = \(\int_0^\infty \big\{1-F(x)-F(-x)\big\} dx\)
b) E(X) = \(\int_{-\infty}^{\infty} \big\{1-F(x)-F(x)\big\} dx\)
c) E(X) = \(\int_0^\infty \big\{1-F(x)-F(x)\big\} dx\)
d) E(X) = \(\int_0^1 \big\{1-F(x)-F(-x)\big\} dx\)
View Answer

Answer: a
Explanation: E(X)=\(\int_{-\infty}^{\infty} x F^{‘}(x)dx = \int_{0}^{\infty} \big\{1-F(x)\big\}dx – \int_{-\infty}^{0}F(x)dx\)
=\(\int_{0}^{\infty} \big\{1-F(x)\big\}dx-\int_0^\infty F(-x)dx\)

4. The second order moment of a random variable X is __________
a) Minimum about the mean of Y
b) Minimum about the mean of X
c) Maximum about the mean of Y
d) Maximum about the mean of X
View Answer

Answer: b
Explanation: E{(X-a)2}
= E{(X-m)2}+(m-a)2
≥ E{(X-m)2},
Since (m-a)2 ≥ 0.

5. For which of the measure of β2, the corresponding density curve has a sharper peak than the normal density curve?
a) If β2<3
b) If β2>3
c) If β2≥3
d) If β2≤3
View Answer

Answer: b
Explanation:\(\beta_2=\frac{\mu_4}{σ^4}\) is called the coefficient of kurtosis. Thus, a high value of β2 is thought to mean a sharply peaked distribution. Now,β2 = \(\frac{3σ^4}{σ^4}\) = 3. So, if β2>3, then the distribution is said to have a sharp peak.

6. The salary of workers in a factory follows a binomial (180, 1/3) distribution. What will be the mean and variance of the distribution curve?
a) 40, 60
b) 40, 50
c) 60, 50
d) 60, 40
View Answer

Answer: d
Explanation: For a binomial (n, p) distribution, the mean is (n*p) and variance is (n*p*(1-p)).
Now, n = 180, p = 1/3, 1-p = 2/3
Mean = (n*p) = 180 * 1/3 = 60
Variance = (60 * 2/3) = 40.

7. For a Binomial (n,p) distribution, variance is _____________
a) n*p
b) n*p*(1-p)
c) n*p*(1-q)
d) n*q
View Answer

Answer: b
Explanation: n is the number of trials; p is the number of successes; q = (1-p) is the number of failures
Hence, the mean is (n*p). Thus, variance is n*p*(1-p).

8. Let X be a Poisson μ variable. Then mean, variance and standard deviation of X are _________
a) μ, μ2, μ3
b) μ, μ , μ
c) μ, μ ,\(\sqrt{\mu}\)
d) \(\sqrt{\mu}\), μ , μ
View Answer

Answer: c
Explanation: For mean X=E(X)=\(\sum_{i=0}^{\infty} + i e^{-\mu} \frac{\mu^{i}}{i!}\)
For variance Var(X)=E{X(X-1)} – m(m-1)
For SD \(\sqrt{Var(x)}\).

9. Standard deviation is the square root of variance.
a) True
b) False
View Answer

Answer: a
Explanation: Variance is used to measure how much spread out the dataset instances are. But, they are calculated in square units. So, standard deviation is introduced to get an idea about the spread in the original scale, by finding the square root of variance.

10. The variance of a random variable X ,Var(X) is defined by ___________
a) Var(X)=E(X2) – {E(X)}2
b) Var(X)=E(X) – {E(X)}
c) Var(X)=E(X2) – {E(X)}
d) Var(X)= {E(X)}2 – E(X2)
View Answer

Answer: a
Explanation: Var(X)=E{(X-m)2}
=E{(X2 – 2Xm + m2)}
=E(X2) – 2E(X).m + m2
=E(X2)-2.m.m + m2
=E(X2) – {E(X)}2.

Sanfoundry Global Education & Learning Series – Probability and Statistics.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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