4 Sums of independent random variables Let 1 Xn independent

4. Sums of independent random variables Let 1, Xn independent random variables and set Y-X1 Xn. (a) Interpret Y and guess its distribution in each of the following cases (at least (i)-(iv)) i. Xin Binom (ni, p), 1 Sisn ii. Xin Pois (Xi), 1 S is n iii. Xi N Neg Binom (ri, p), 1 Sis n G. Fellouris STAT 400 FALL 2015 iv. Xin Earp A), 1 sis n (b) Compute the migf of Y and find its distribution in each of the above cases.

Solution

Given X₁, X₂, X₃ ,….. X_n, are independent random variables and Y= X₁, X₂,…. X_n, that implies Y= X_i

Let us consider for X₁ binomial(n₁,p) X₂ binomial(n₂,p) be independent random variables then X₁ +X₂ binomial(n₁₊n₂,p)

According to the property of moment generating function

If X₁, X₂, X₃ ,….. X_n, are independent random variables then M_{x1+x2+……..xn}(t) = M_x1(t) M_x2(t)……….M_xn(t)

X_i binomial(n_i,p) for all i=1,2,….n

Let us consider for X₁ pois(_i) X₂ pois(_i) be independent random variables then X₁ +X₂ pois(₁+_i) According to the property of moment generating function

If X₁, X₂, X₃ ,….. X_n, are independent random variables then M_{x1+x2+……..xn}(t) = M_x1(t) M_x2(t)……….M_xn(t)

X_i pois(₁ +₂+…. +_n ) for all i=1,2,….n

The moment generating function is M_x(t) = p^r(1-qe^t)^-r

and M_x1+x2(t) = M_x1(t) M_x2(t)

If X₁, X₂, X₃ ,….. X_n, are independent random variables then M_{x1+x2+……..xn}(t) = M_x1(t) M_x2(t)……….M_xn(t)

X_i neg bionor(r₁r₂……..r_n, p)for all i=1,2,….n

For this distribution we can’t interpret Y

For this distribution we can’t interpret Y