Conditional Distribution Of One Variable Given Another In A Multinomial Distribution

Suppose X_{1}, X_{2,} X_{3} ~ Multinomial(n, $\displaystyle \theta$_{1}, $\displaystyle \theta$_{2}, $\displaystyle \theta$_{3}).

I know that X_{1}, X_{2,} X_{3} are individually Binomial(n, $\displaystyle \theta$_{1}), Binomial(n, $\displaystyle \theta$_{2}), Binomial(n, $\displaystyle \theta$_{3}) respectively.

I'd like to know if:

1) X_{1}, X_{2,} X_{3} are independent

2) If they are, whether that means that the conditional distribution of X_{2} given X_{1} would be simply Binomial(n, $\displaystyle \theta$_{2})?

Thanks.

Re: Conditional Distribution Of One Variable Given Another In A Multinomial Distribut

Hey JohnDoe2013.

The easiest way to answer this is to look at the joint PDF: If it can be factored then you have independence.

As an example, you should have P(A = a, B = b, C = c) = P(A=a)*P(B=b)*P(C=c)

Re: Conditional Distribution Of One Variable Given Another In A Multinomial Distribut

Hi Chiro,

This is a strange one. The definition of the Multinomial Distribution says that the random variables are independent and identically distributed.

However, since we are sampling a fixed number of times say n, we have $\displaystyle X_1+X_2+X_3=n$ which means they are not independent.

My thoughts are then that:

P($\displaystyle X_2=x_2 | X_1=x_1$) = $\displaystyle n-x_1 \choose x_2$$\displaystyle \theta_2^{x_2}$$\displaystyle (1-\theta_2)^{n-x_1-x_2}$

Would that be correct?

Thanks.

Re: Conditional Distribution Of One Variable Given Another In A Multinomial Distribut

If the X's refer to counts in the multinomial then yes they are not independent.

Defining X1 will define X2 and defining X2 will define X3. Independence requires that any value of X1 doesn't probabilistically affect X2 (and same for X2 and X3).

In other words, P(A|B) = P(A) for independence and clearly P(A|B) is not only a function of A but also B as well.