Question

Consider the below integrals and series. First, attempt to generate a pdf or a pmf from each of them and justify why if you can't. Second, if you successfully generated a pdf or a pmf in the first, find the named distribution which your pdf/pmf belongs to (need not be in lecture notes) Σ +r-lc, pェ 1 -, r is a given positive integer and p is probability value. = (1-p) k=0 2. z"-i exp (-z)dz = rn, n e [1, oc), and rr denotes the ganma function. 5.

Answer 1

intger and p is pobability value Thus, x+r-, Cx Pt CI-P) is valid pm because x:o n+1 n "Х is nt pm4because In(lt)

k-1 othe rwi becau) Now 1 χ denotes -the gamma funtion x : exp(-x) dx = 1 since h valid pdt.

and this is gamma distibon with paramet 1 and n ie. cwith single para meir 'n' ~ Gamma (1,n), and pdf is, ←02 す-S tett +('t ) İS ·ua/id pd t s-vd because, f/X)>0 an star da-d Cauchy disMbcton (*) 75 and polf