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

1. The mean and the variance for gamma distribution are __________

a) E(X) = 1/λ, Var(X) = α/λ^{2}

b) E(X) = α/λ, Var(X) = 1/λ^{2}

c) E(X) = α/λ, Var(X) = α/λ^{2}

d) E(X) = αλ, Var(X) = αλ^{2}

View Answer

Explanation: The mean and the variance for gamma distribution is given as

E(X) = α/λ, Var(X) = 1/λ

^{2}.

2. Putting α=1 in Gamma distribution results in _______

a) Exponential Distribution

b) Normal Distribution

c) Poisson Distribution

d) Binomial Distribution

View Answer

Explanation: f (x) = λ

^{α}x

^{α−1}e

^{−λx}/ Γ(α) for x > 0

= 0 otherwise

If we let α=1, we obtain

f(x) = λe

^{−λx}for x > 0

= 0 otherwise.

Hence we obtain Exponential Distribution.

3. Sum of n independent Exponential random variables (λ) results in __________

a) Uniform random variable

b) Binomial random variable

c) Gamma random variable

d) Normal random variable

View Answer

Explanation: Gamma (1,λ) = Exponential (λ).

Hence Exponential (λ

_{1}) + Exponential (λ

_{2})+ Exponential (λ

_{3})….. n times = Gamma(n,λ).

4. Find the value of Γ(5/2).

a) 5/4 . π^{1/2}

b) 7/4 . π^{1/2}

c) 1/4 . π^{1/2}

d) 3/4 . π^{1/2}

View Answer

Explanation: By the property of Gamma Function

Γ(α+1) = αΓ(α)

\(Γ(\frac{5}{2}) = \frac{3}{2} ⋅ Γ(\frac{3}{2}) \)

\(= \frac{3}{2} ⋅ \frac{1}{2} . Γ(\frac{1}{2}) \)

\(= \frac{3}{2} ⋅ \frac{1}{2} ⋅ π^{1/2} \) By property of Gamma function \(Γ(\frac{1}{2}) = π^{1/2} \)

\(= \frac{3}{4} . π^{1/2}. \)

5. Gamma function is defined as Γ(α) = _{0}∫^{∞} x^{α−1} e^{−x} dx.

a) True

b) False

View Answer

Explanation: The Gamma function is defined as Γ(α) =

_{0}∫

^{∞}x

^{α−1}e

^{−x}dx. Gamma function can also be defined as Γ(α+1) = αΓ(α).

6. Gamma distribution is Multi-variate distribution.

a) True

b) False

View Answer

Explanation: Gamma distribution is a uni-variate distribution that means it is only defined for x ranging from (0, ∞).

7. Which of the following graph represents gamma distribution?

a)

b)

c)

d)

View Answer

Explanation: Gamma distribution is defined as

f(x) = λ

^{α}x

^{α−1}e

^{−λx}/ Γ(α) for x > 0.

Hence it is an exponentially decreasing function.

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

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