Cumulative distribution function - Poisson distribution:
Find the cdf of Y = X1, + X2 where XI and X2 independently follow Poisson distribution with parameters λl and λ2 respectively.
Solution:
The cdf of Y is
F(y) =P(Y≤ y) =P (X1 + X2 ≤ y)
= ∑ ∑ e - λ1 λ1 x1/x1! e- λ2 λ2x2
x1+x2 ≤ y
The summation is taken over all integer values of ( xl, x2 ) satisfying xl ≥ 0, x2 ≥ 0, x1 + x2 ≤ y . Making the substitution r = xl + x2, x1 = x We have xl = x, x2, = r-x.
Thus, region of summation in terms of r, x is all integers (x, r ) satisfying
x ≥ 0 , r - x ≥ 0, r ≤ y
or
0 ≤ x ≤ r ≤ y
Writing the summation in terms oi r and x, rearranging the terms, we have
where λ = λ l + λ2 . The final expression of F (y) is the cdf of Poisson distribution with parameter λ. Hence Y also follows a Poisson distribution with parameter λ.