Question

X~ bin(n,p)
and$X_2\sim bin(n,p)$ are independent what is the distribution of Y=X1+X2

Answer 1

TOPIC:Binomial distribution.

=) Griven that, EX, ~ Bin (m, b). Y independently. (x2 ~ Bin (n, b) " so, the pony of x, is- P(x1 - x) = { m ) p q n-4 ; don, 24 = 0,1,--, n. ; elsewhere. - where, q =(1-7). Now, the mgf of X, is - = Mx, (+) = E(etxi). = F ett p(x, = х4). 240 Keo - Catbln

Similarly, the mgf of X2 is Mez (t) = (a the ")" I As, x, and Xz have the same distribution. =) So, the ings of y(x, + X) is - - Mxitx (t) = E(e + (x + x2) (Due to independence a Eletxi). E (eng. - Mx, (+). Mx (+). = (at het)" (athen = (a + bety an - which is the mgf of Bin (2n, 6) - re m - distribution. :. By the uniqueness of mas; we can conclude that, Ly=(x, + X2} ~ Bin (2n, p) Ang