# 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

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

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) = $$\int_0^1 \big\{1-F(x)-F(-x)\big\} dx$$

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

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

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

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

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

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

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

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)

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)-m2
=E(X2) – {E(X)}2.

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