Gamma distribution has a lot of special cases such as expone

Gamma distribution has a lot of special cases (such as exponential and chi-square). If X follows gamma(? = 3 2 , ? = 1) (parameterized such that E(X) = ??), then Y = ? X follows a Maxwell distribution. The probability density function (pdf) of Y is given by in the picture.

a) Calculate E(Y ).

b) Calculate V ar(Y ).

c) Another special case of the gamma distribution is given by: if W follows exponential(? = 1), then V = ? 2W follows a Rayleigh distribution. Derive the probability density function (pdf) of V .

Problem 3. (Re-do this problem from MidtermII) Gamma distribution has a lot of special cases (such as exponential and chi-square). IEX follows gamma(-3,A-1) (parameterized such that E(X)-), then Y-yx follows a Maxwell distribution. The probability density function (pdf) of Y is given by a) Calculate E(Y) b) Calculate Var() c) Another special case of the gamma distribution is given by: if W follows exponential( then V of V. ) V2W follows a Rayleigh distribution. Derive the probability density function (pdf)

Solution