Is X1 Given S=s a Binomial Distribution in Poisson Variables?

[johnnytzf]

let X1 and X2 be independent Poisson variables with respective parameters μ1 and μ2. Let S = X1 + X2. Is X1 given S=s a binomial dsitribution? What is the parameters?


I just can show that S is a Poisson with mean μ1 + μ2. But I am not confirm X1 given S is a binomial or not? Someone please help to prove it.

 

[mathman]

S cannot be binomial. X2 ranges over all integers. Therefore S will also, while binomial range is finite.

 

[statdad]

If the conditional distribution is intended you want to know about X1 when S = X1 + X2 = s, a fixed value. In this setting X1 cannot range over all integer values.

 

[mathman]

If the conditional distribution is intended you want to know about X1 when S = X1 + X2 = s, a fixed value. In this setting X1 cannot range over all integer values.

You're right. I misread the question.

It looks binomial:

P(X1=x|S=s) = (μ1^xμ2^(s-x))/{x!(s-x)!(μ1+μ2)^s}

 

[mathman]

You're right. I misread the question.

It looks binomial:

P(X1=x|S=s) = (μ1^xμ2^(s-x))/{x!(s-x)!(μ1+μ2)^s}

I was sloppy. There should be a coefficient of s!