If X has the Gamma distribution with parameters α and β; i.e., if its p.d.f. is given by

x > 0 (and0 for x ‰¤ 0),

Where the Gamma function ᴦ (α) is given

Then show that

fx(t) = 1/(1 €“ iβt) α

In particular, for α = 1 and β = 1/λ, we get the ch.f. of the Negative Exponential distribution with parameter λ; i.e., fx(t) = 1/(1 it/λ) and for α = r/2 (r > 0 integer) and β = 2, we get the ch.f. of the chi-square distribution with r degrees of freedom; i.e.,

fx(t) = 1/(1 - 2it)^r/2.

Transcribed Image Text:

p(x; α, β) - xα-lea/β, Г'а)ва ito