Cumulative distribution function - Poisson distribution:
Find the cdf of Y = X_{1}, + X_{2} where X_{I} and X_{2} independently follow Poisson distribution with parameters λ_{l} and λ_{2} respectively.
Solution:
The cdf of Y is
F(y) =P(Y≤ y) =P (X_{1} + X_{2} ≤ y)
= ∑ ∑ e ^{-}^{ λ1} λ_{1} ^{x1}/x1! e^{- λ2} λ_{2}^{x2}
x_{1}+x_{2} ≤ y
The summation is taken over all integer values of ( x_{l}, x_{2} ) satisfying x_{l} ≥ 0, x_{2} ≥ 0, x_{1 }+ x_{2} ≤ y . Making the substitution r = x_{l} + x_{2}, x_{1} = x We have x_{l} = x, x_{2}, = 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} + λ_{2} . 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.