PDF of Gamma (Γ) Distribution for α = 1.5, k = 5 & x = 15
Gamma (Γ) distribution example problem workout with steps & calculation summary for the probabilty density function (PDF) of the random variable x = 15, scale parameter k = 5 and shape parameter α = 1.5 in statistical experiments.
| Calculation Summary | |
|---|---|
| Shape Parameter (α) | 1.5 |
| Scale Parameter (k) | 5 |
| random variable (x) | 15 |
| 0.0195 | |
| mean (µ) | 7.5 |
work with steps for α = 1.5, k = 5 & x = 15
Question:
A random variable x = 15 follows the gamma distribution which has the shape parameter α = 1.5 and scale parameter k = 5. Find the probability density function P(x) for the random variable x?
workout :A random variable x = 15 follows the gamma distribution which has the shape parameter α = 1.5 and scale parameter k = 5. Find the probability density function P(x) for the random variable x?
step 1 Addres the formula input parameter and values
shape parameter α = 1.5
scale parameter k = 5
x = 15
step 2 Find probability density function using k, α & x values
PDF = (x)(α - 1) e -(x/k)Γ(α) kα
= (15)(1.5 - 1) e -(15/5)Γ(1.5) 51.5
= (15)(0.5) e -(3)Γ(1.5) (5)1.5
0.8862 is the gamma value of 1.5
= (3.8729833462074) (0.0498)(0.8862) (11.1803)
= 0.1928/9.908
PDF = 0.0195
step 3 Find Mean value using shape & scale parameter
Mean µ = k x α
= 5 x 1.5
Mean µ = 7.5
