Let X_1, X_2 ....,be a sequence of nonegative i.i.d. random variables with common mean μ_1and variance (σ_1)^2 αnd let Y_1,Y_2....,be another sequence of nonnegative i.i.d. random variables with common mean μ_2 and (σ_2)^2, Let N be a counting random variable independent of X's and Y's.
Prove that corr[(summation of X_i where i runs from 1 to X), N]<=corr[(summation of Y_i where i runs from1 to N),N] is equivalent to cv(X)>=cv(Y),where cv(X)=(σ_1)^2/(μ_1)^2 is the coefficient of variation of X. Similarly, we define cv(Y).
Now apply this result to the case when (X_i)'s have Gamma distribution with parameters (α_1,β_1) and (Y_i)'s have Gamma distribution with parameters (α_2, β_2),respectively and find a necessary and sufficient condition on the parameters for corr[(summation of X_i where i runs from 1 to X), N]<=corr[(summation of Y_i where i runs from1 to N),N] to hold.