identically distributed random variables

identically distributed random variables

Let Z = X₁ + · · · + X_r. where the X/s are r independent, identically distributed random variables, each having an exponential distribution with the (same) parameter a. Then Z has a Gamma distribution with parameters a and r.

 Notes:

(a) Is not true if the parameters of the various exponential distributions are different. This becomes evident when we consider the mgf of the resulting sum of random variables.
(b) The following corollary of the above theorem has considerable importance in certain statistical applications: The random variable W = 2αZ has distribution X²_2r

· This is an immediate consequence of the fact that M_w(t) = M_z(2αt) = [α/(α - 2αt)]^r = (1 - 2t)^-2rl2   yields the above corollary. Thus we may use the tabulated chi-square distribution in order to evaluate certain probabilities associated with Z. For example, P(Z ≤ 3) = P(2αZ ≤ 6α). This latter probability may be obtained directly from the tables of the chi-square distribution, if a and r are given.