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

1. In a Binomial Distribution, if ‘n’ is the number of trials and ‘p’ is the probability of success, then the mean value is given by ___________

a) np

b) n

c) p

d) np(1-p)

View Answer

Explanation: For a discrete probability function, the mean value or the expected value is given by

Mean (μ)=\(\sum_{x=0}^{n}xp(x)\)

For Binomial Distribution P(x)=

^{n}C

_{x}p

^{x}q

^{(n-x)}, substitute in above equation and solve to get

µ = np.

2. In a Binomial Distribution, if p, q and n are probability of success, failure and number of trials respectively then variance is given by ___________

a) np

b) npq

c) np^{2}q

d) npq^{2}

View Answer

Explanation: For a discrete probability function, the variance is given by

Variance (V)=\(\sum_{x=0}^n x^2p(x)-\mu^2\)

Where µ is the mean, substitute P(x)=

^{n}C

_{x}p

^{x}q

^{(n-x)}in the above equation and put µ = np to obtain

V = npq.

3. If ‘X’ is a random variable, taking values ‘x’, probability of success and failure being ‘p’ and ‘q’ respectively and ‘n’ trials being conducted, then what is the probability that ‘X’ takes values ‘x’? Use Binomial Distribution

a) P(X = x) = ^{n}C_{x} p^{x} q^{x}

b) P(X = x) = ^{n}C_{x} p^{x} q^{(n-x)}

c) P(X = x) = ^{x}C_{n} q^{x} p^{(n-x)}

d) P(x = x) = ^{x}C_{n} p^{n} q^{x}

View Answer

Explanation: It is the formula for Binomial Distribution that is asked here which is given by P(X = x) =

^{n}C

_{x}p

^{x}q

^{(n-x)}.

4. If ‘p’, ‘q’ and ‘n’ are probability pf success, failure and number of trials respectively in a Binomial Distribution, what is its Standard Deviation?

a) \(\sqrt{np}\)

b) \(\sqrt{pq}\)

c) (np)^{2}

d) \(\sqrt{npq}\)

View Answer

Explanation: The variance (V) for a Binomial Distribution is given by V = npq

Standard Deviation = \(\sqrt{variance} = \sqrt{npq}\).

5. In a Binomial Distribution, the mean and variance are equal.

a) True

b) False

View Answer

Explanation: Mean = np

Variance = npq

∴ Mean and Variance are not equal.

6. It is suitable to use Binomial Distribution only for ___________

a) Large values of ‘n’

b) Fractional values of ‘n’

c) Small values of ‘n’

d) Any value of ‘n’

View Answer

Explanation: As the value of ‘n’ increases, it becomes difficult and tedious to calculate the value of

^{n}C

_{x}.

7. For larger values of ‘n’, Binomial Distribution ___________

a) loses its discreteness

b) tends to Poisson Distribution

c) stays as it is

d) gives oscillatory values

View Answer

Explanation: P(x)=\(\lim_{n\rightarrow\infty}c_x^n p^x q^{n-x}=\frac{e^{-m}m^x}{x!}\)

Where m = np is the mean of Poisson Distribution.

8. In a Binomial Distribution, if p = q, then P(X = x) is given by?

a) ^{n}C_{x} (0.5)^{n}

b) ^{n}C_{n} (0.5)^{n}

c) ^{n}C_{x} p^{(n-x)}

d) ^{n}C_{n} p^{(n-x)}

View Answer

Explanation: If p = q, then p = 0.5

Substituting in P(x)=

^{n}C

_{x}p

^{x}q

^{(n-x)}we get

^{n}C

_{n}(0.5)

^{n}.

9. Binomial Distribution is a ___________

a) Continuous distribution

b) Discrete distribution

c) Irregular distribution

d) Not a Probability distribution

View Answer

Explanation: It is applied to a discrete Random variable, hence it is a discrete distribution.

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