# PDF of Gamma (Γ) Distribution for α = 9, k = 3 & x = 11

Gamma (Γ) distribution example problem workout with steps & calculation summary for the probabilty density function (PDF) of the random variable x = 11, scale parameter k = 3 and shape parameter α = 9 in statistical experiments.

Calculation Summary
Shape Parameter (α) 9
Scale Parameter (k)3
random variable (x)11
PDF 0.0069
mean (µ)27

## work with steps for α = 9, k = 3 & x = 11

Question:
A gamma distribution which has the shape parameter α = 9 and scale parameter k = 3. Find the probability density function P(x) for the random variable x = 11?
workout :
step 1 Addres the formula input parameter and values
shape parameter α = 9
scale parameter k = 3
x = 11

step 2 Find probability density function using k, α & x values
PDF = (x)(α - 1) e -(x/k)Γ(α) kα

= (11)(9 - 1) e -(11/3)Γ(9) 39

= (11)(8) e -(3.6667)Γ(9) (3)9

40320 is the gamma value of 9

= (214358881) (0.0256)(40320) (19683)

= 5479341.6548/793618560

PDF = 0.0069

step 3 Find Mean value using shape & scale parameter
Mean µ = k x α
= 3 x 9
Mean µ = 27