Question

At a certain farm, the number of eggs that hatch each day, X, is a Poisson(μ) random variable, while the number of eggs that do not hatch, Y, is an independent Poisson(ν) random variable. Suppose it is known that there were n total eggs one day. What is the conditional distribution of the number of eggs that hatch?

Answer 1

We are given following data :

• No. of eggs that hatch each day , X ~ Poi ( )
• No. of eggs that do not hatch , Y ~ Poi ( )
• X and Y are independent of each other ,   X + Y ~ Poi ( + )
• There were total n eggs on one day , X + Y = n

We want the Conditional Distribution of number of eggs that hatch each day i.e. , ( X | X + Y ).

Let No. of eggs that hatch one day equal to r .

X is a Poisson R.V. with PDF :

Y is a Poisson R.V. with PDF :  

X + Y is a Poisson R.V. with PDF :  

Therefore , Conditional Probability is computed as follows :

......... Since : P ( A | B ) = P ( A B ) / P ( B )

...................... Since , X and Y are independent

Substituting the value of Probabilities :

........... Add and Subtract r in power of  

This resembles the PDF of Binomial ( n , p ) Distribution i.e. ,

with :

Therefore :