Let Y1 Y2 Yn denote a random sample from a Poisson distribu

Let Y1, Y2, ..., Yn denote a random sample from a Poisson distribution with parameter ?. Show by conditioning that ? i=1 to n Y_i is sufficient for ?.

Solution

First let us prove for y = y1+y2. Then extend for n numbers.

Sum of Poisson Random Variables. Let Y1 and Y2 be independent Poisson random variables where Xi has a Poisson (?i) distribution for i = 1, 2.

Then Y1 + Y2 has a Poisson distribution with ?1 + ?2

Extend to i = n

Let Y1, Y2, ...Yk be independent Poisson random variables where Yi has a Poisson (?i) distribution for i = 1, 2, ..., k. Then Y1 + Y2 +