Please show the full steps. The gamma distribution is a family of continuous probability distributions withpositive parameters r, A and a density given byx"-le-Ax_I(r)f(x) =when x > 0, and zero otherwise.||Here I(r) is called the gamma function and defined by T(r) = S x"-le¬®dx. Seehttps://en.wikipedia.org/wiki/Gamma.function for more details.Say X -on the properties of gamma distributions.gamma(r, X) if X has the density above. The following problems are The gamma function is a fundamental function in mathematics. Oneof its many amazing properties is thatГ(п) — (п — 1)!, for n %3= 1,2, ....Thus the gamma function is a generalization of the factorials. Let us prove thisidentity using probability following these steps.(i) Take r = n, 1 = 1 and find E(X) either using problem 2 or problem 3.(ii) Show that E(X) is also the ratio I(n+ 1)/T(n).(iii) Use induction now to complete the argument.

Name: Please show the full steps. The gamma distribution is a family of cont
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please show the full steps. The gamma distribution is a family of continuous probability distributions withpositive parameters r, A and a density given byx"-le-Ax_I(r)f(x) =when x > 0, and zero otherwise.||Here I(r) is called the gamma function and d