# Probability and Statistics Questions and Answers – Gamma Distribution

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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/λ2
b) E(X) = α/λ, Var(X) = 1/λ2
c) E(X) = α/λ, Var(X) = α/λ2
d) E(X) = αλ, Var(X) = αλ2

Explanation: The mean and the variance for gamma distribution is given as
E(X) = α/λ, Var(X) = 1/λ2.

2. Putting α=1 in Gamma distribution results in _______
a) Exponential Distribution
b) Normal Distribution
c) Poisson Distribution
d) Binomial Distribution

Explanation: f (x) = λα xα−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

Explanation: Gamma (1,λ) = Exponential (λ).
Hence Exponential (λ1) + Exponential (λ2) + Exponential (λ3)….. n times = Gamma(n,λ).

4. Find the value of Γ(5/2).
a) 5/4 . π1/2
b) 7/4 . π1/2
c) 1/4 . π1/2
d) 3/4 . π1/2

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 xα−1 e−x dx.
a) True
b) False

Explanation: The Gamma function is defined as Γ(α) = 0 xα−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

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)

Explanation: Gamma distribution is defined as
f(x) = λα xα−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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