Conditional Distribution of Multinomial Random Variables

[broegger]

I've been staring at this for hours. Any hints?

Let the vector $$Y = (Y_1,Y_2,\dots,Y_k)$$ have a multinomial distribution with parameters n and $$\pi = (\pi_1,\pi_2,\dots,\pi_k)$$:

$$\sum_{i=1}^{k}Y_i = n, \quad \sum_{i=1}^{k}\pi_i = 1$$​

Show that the conditional distribution of $$Y_1$$ given $$Y_1+Y_2=m$$ is binomial with n = m and $$\pi = \frac{\pi_1}{\pi_1+\pi_2}$$.

I've tried to apply the definition of a conditional probability and sum over the relevant events in the multinomial distribution, but it gives me nothing.

Thanks.

 

[ZioX]

Hmmm, this works if they are Poisson. Not sure if it works if it is multinomial. The standard way to do this would be to compute P[X_1=x|X_1+Y_2=m], as you tried.

Edit: I should be more precise. If Y_1 and Y_2 poisson r.v. with paramaters lambda1 and lambda2, then Y_1 | Y_1+Y_2=m is distributed as Binomial(m, lambda1/(lambda1+lambda2)).