Gamma Distribution MCQs

Correct Answer: c) 18

Explanation:
For a Gamma distribution with MGF (1−θt)^{-α}, we have:
Mean = αθ
Variance = αθ²
Here MGF = (1−3t)^{-2}, so α = 2 and θ = 3
Variance = αθ² = 2×3² = 2×9 = 18

Correct Answer: c) 5

Explanation:
For a Gamma distribution:
Mean = αθ and Variance = αθ²
Given: Mean = 20, Variance = 80
So: αθ = 20 and αθ² = 80
Dividing: (αθ²)/(αθ) = θ = 80/20 = 4
Substituting: α×4 = 20 ⇒ α = 5

Correct Answer: c) 6

Explanation:
For a Gamma distribution with shape parameter α and scale parameter θ:
Mean = αθ
Given: α = 3, θ = 2
Mean = 3×2 = 6

Correct Answer: d) Skewness

Explanation:
The shape parameter (α) primarily determines the skewness of the gamma distribution.
For a gamma distribution, skewness = 2/√α
As α increases, the distribution becomes more symmetric (less skewed).
Both mean and variance depend on both shape and scale parameters.

Correct Answer: d) 9

Explanation:
For a Gamma distribution with shape parameter k and scale parameter θ:
Variance = kθ²
Given: k = 4, θ = 1.5
Variance = 4×(1.5)² = 4×2.25 = 9

Correct Answer: a) (β^α x^(α-1) e^(-βx))/Γ(α)

Explanation:
The PDF of a Gamma distribution with shape parameter α and rate parameter β is:
f(x) = (β^α x^(α-1) e^(-βx))/Γ(α) for x > 0
where Γ(α) is the gamma function.
Option b is not a valid PDF.
Option c is the PDF of a standard normal distribution.
Option d is not a valid PDF either.

Correct Answer: b) αθ

Explanation:
For a Gamma distribution with shape parameter α and scale parameter θ:
Mean = αθ
This can be derived from the moment generating function or directly from the definition of expected value using the PDF.

Correct Answer: b) (1-βt)^(-α)

Explanation:
The MGF of a Gamma distribution with shape parameter α and rate parameter β is:
M(t) = (1-βt)^(-α) for t < 1/β
Note: If using scale parameter θ (where θ = 1/β), then MGF = (1-θt)^(-α)

Correct Answer: a) 0.0070

Explanation:
With Poisson arrivals (1 per minute), waiting time for n customers follows Gamma(n, λ) where λ = 1/minute.
For the first 2 customers: X ~ Gamma(2, 1)
P(X > 5) = 1 - P(X ≤ 5)
For Gamma(2,1): P(X ≤ x) = 1 - e^(-x) - xe^(-x)
P(X ≤ 5) = 1 - e^(-5) - 5e^(-5) = 1 - 6e^(-5) ≈ 0.993
P(X > 5) = 1 - 0.993 ≈ 0.007

Correct Answer: a) 7, 2

Explanation:
When individual lifetimes follow Exp(λ), the time until the kth failure follows Gamma(k, λ).
Given: Each machine has exponential lifespan with mean 0.5 year, so rate λ = 1/0.5 = 2
For time until 7th failure: X ~ Gamma(7, 2)
So shape = 7 and rate = 2

Correct Answer: c) It is used to model waiting times

Explanation:
a) False: Gamma distribution is defined for x > 0 only.
b) False: Gamma distribution is generally right-skewed, becoming more symmetric as shape parameter increases.
c) True: Gamma distribution commonly models waiting times, especially for multiple events in a Poisson process.
d) False: Gamma distribution is continuous, not discrete.

Correct Answer: c) √20

Explanation:
For Gamma(α, θ) with shape α and scale θ:
Variance = αθ²
Given: α = 5, θ = 2
Variance = 5×2² = 5×4 = 20
Standard deviation = √Variance = √20 ≈ 4.47

Correct Answer: a) Shape parameter = 1

Explanation:
The exponential distribution with rate λ is equivalent to a Gamma distribution with shape α = 1 and rate β = λ (or scale θ = 1/λ).
When shape parameter = 1, the PDF of Gamma becomes f(x) = λe^(-λx) for x > 0, which is exactly the PDF of an exponential distribution.

Correct Answer: c) αθ²

Explanation:
For a Gamma distribution with shape parameter α and scale parameter θ:
Mean = αθ
Variance = αθ²
This formula can be derived from the moment generating function or using properties of the gamma function.

Correct Answer: a) 0.2642

Explanation:
For Gamma(2,1), the CDF is:
F(x) = 1 - e^(-x) - xe^(-x)
P(X<1) = F(1) = 1 - e^(-1) - 1×e^(-1)
= 1 - e^(-1)(1+1)
= 1 - 2e^(-1)
= 1 - 2/e ≈ 0.2642

Correct Answer: b) It only exists for t<1/β

Explanation:
The MGF of a Gamma distribution with shape α and rate β is:
M(t) = (1-βt)^(-α)
This expression is only defined when 1-βt > 0, which means t < 1/β.
For values of t ≥ 1/β, the integral in the MGF definition diverges.

Correct Answer: c) Taking the nth derivative of the MGF at t = 0

Explanation:
The nth moment about the origin (E[X^n]) can be found by:
E[X^n] = M^(n)(0)
Where M^(n)(t) is the nth derivative of the MGF with respect to t.
This is a general property of MGFs, not just for gamma distributions.

Correct Answer: c) Both shape and scale

Explanation:
The shape parameter (α) primarily affects the skewness and overall shape of the distribution.
The scale parameter (θ) stretches or compresses the distribution along the x-axis.
Together, they determine the complete shape of the gamma distribution curve.
Different combinations of these parameters result in differently shaped gamma distributions.

Correct Answer: c) Exponential

Explanation:
The gamma distribution generalizes the exponential distribution.
The exponential distribution is a special case of the gamma distribution with shape parameter α = 1.
The gamma distribution can be thought of as the sum of α independent exponential random variables.

Correct Answer: a) 6

Explanation:
The MGF (1−2t)^(-3) corresponds to a gamma distribution with parameters α = 3 and θ = 2
(where MGF form is (1−θt)^(-α))
For a gamma distribution: Mean = αθ = 3×2 = 6