Splitting a Poisson random variable. A transmitter sends

QUESTION:

Splitting a Poisson random variable. A transmitter sends out either a 1 with probability p, or a 0 with probability 1 - p. independent of earlier transmissions. If the number of transmissions within a given time interval has a Poisson PlF with parameter A, show that the number of Is transmitted in that same time interval has a Poisson PMF with parameter pA. Solution. Let X and Y be the numbers of Is and Os transmitted, respectively. Let Z = X + Y be the total number of symbols transmitted. We have Thus, P(X = n, Y = m) = P(X = n, Y = m I Z = n + m)P(Z = n + m) = (n + m) p n( 1 _ p)111 . e-'>' A n+111 n (n + m)! e-'>'P(Ap)n e-.>.( l -p) (A(l _ p))111 n! m! oc P(X = n) = L P(X = n, Y = m) '>'P( ' ) n x (A(1 )) 111 = e p e-.>.(l-p) '"' -p n. m! 111=0 -'>'P( ' ) n _ e P - .>. (l-p) '>'(l - -p) ,e e n. e-'>'P(Ap)n n! so that X is Poisson with parameter Ap.