2.4 The Gamma Distribution The gamma distribution has a pdf of the form fit) —r(a)A(2trie4t.
Or
t > 0
f(t) /re-/eh t>0 r(a) where a > 0 and A > 0 are parameters; A is a scale parameter and a is sometimes called the index or shape parameter. This distribution, like the %‘’ eitmll distribution, includes the exponential as a special case (a = I).
The distribution function is F(t) = J f(u)du.
When a > 0 ,
aacitY F(0- / 7—e u, t > 0 1-0 a! a-t(Aty F(0= e-Ar t > 0 t-o r. The gamma distribution is DFR for 0 < a < I and WA for a > 1. For a = 1, it has a constant failure rate. The hazard function is kW= f At). It can be shown to be monotone increasing for a > 1, with h(0) = 0 and lime h(t)= A . For 0 < a < I h(t) is monotone decreasing. with lime-0 h(t) = 00 and limr_,„ h(t) = A .
For 0 <a < ),(negative aging). hazard rate decreases monotonically from infinity to A as time increases.
For a > 1. (positive aging), hazard rate increases monotonically from zero to A as time increases. For a = I, hazard rate equals A , a constant. The gamma distribution describes a different type of survival pattern where the hazard rate is decreasing or increasing to a constant value as time approaches infinity.

For Gamma Distribution of Time (t) find clearly Math Proof for each of that :

1) f(t) ? the pdf of gamma distribution of t .
2) F(t) ? the cdf of gamma distribution of t .
3) S(t)? the survival function of gamma distribution of t .
4) h(t)? the hazard function of gamma distribution of t.
5)MTTF ? Mean Time to Failure for Gamma distribution of t.