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
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
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
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
View Answer
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
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
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
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
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
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
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.
Sanfoundry Global Education & Learning Series – Probability and Statistics.
To practice all areas of Probability and Statistics, here is complete set of 1000+ Multiple Choice Questions and Answers.
If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]
- Practice Numerical Methods MCQ
- Practice Engineering Mathematics MCQ
- Check Probability and Statistics Books
- Apply for 1st Year Engineering Internship
