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Chapter 5: Q. 5.22 (page 216)

Compute the hazard rate function of a gamma random variable with parameters (α,λ)and show it is increasing when α≥1and decreasing when α≤1

Short Answer

Expert verified

That is, t1≤t2implies λXt1≤λXt2when α-1≥0.

λX(t)is increasing when α≥1similarly, λX(t)is decreasing when α≤1

Step by step solution

01

Determine the Function of the hazard.

Let Xbe a gamma random variable with parameters (α,λ)

Thenf(x)=λe-λx(λx)α-1Γ(α)x≥0,x<0

02

Simplify the value.

We can simplify it, then we have

λX(t)=1∫t∞e-λ(x-t)xtα-1dx

Let y=x-t,

λX(t)=1∫0∞e-λy1+ytα-1dy

If α-1≥0,for t1≤t2,y>0

We have,

Hence,

λXt1=1∫0∞e-λy1+yt1α-1dy
≤1∫0∞e-λy1+yt2α-1dy=λXt2λXt1=1∫0∞e-λy1+yt1α-1dy

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