Conditional Expectation problem

**downthesun01** · August 3rd 2011, 08:28 PM

Consider n multinomial trials, where each trial independently results in outcome i with probability $p_i,\sum_{i=1}^{k}p_{i}=1$ . With $X_i$ equal to the number of trials that result in outcome i, find $E[X_{1}|X_2>0].$

I'm having a hard time understanding what the question is asking. Am I looking for the average number of trials that result in $X_1$ given that there's at least one trial that results in $X_2$ ?

EDIT:

After thinking about it, isn't the answer just

$1-n\sum^{k}_{i=2}p_i$

I don't know, I don't think I'm getting the question.

**CaptainBlack** · August 14th 2011, 10:52 PM

Originally Posted by downthesun01

Consider n multinomial trials, where each trial independently results in outcome i with probability $p_i,\sum_{i=1}^{k}p_{i}=1$ . With $X_i$ equal to the number of trials that result in outcome i, find $E[X_{1}|X_2>0].$

I'm having a hard time understanding what the question is asking. Am I looking for the average number of trials that result in $X_1$ given that there's at least one trial that results in $X_2$ ?

To work out the conditional expectation you need the conditional probability:

EDIT:

After thinking about it, isn't the answer just

$1-n\sum^{k}_{i=2}p_i$

I don't know, I don't think I'm getting the question.

$P(X_1=x_1|X_2>0)+P(X_1=x_1|X_2=0)=P(X_1=x_1$

So:

$P(X_1=x_1|X_2>0)=P(X_1=x_1)-P(X_1=x_1|X_2=0)$

Now the distribution of $X_1$ conditioned on $X_2=0$ is binomial with number of trials equal to $n$ and probability of a success being $p_1^*=p_1/(p_1+p_3+...+p_k)=p_1/(1-p_2)$ , and the unconditional probability is binomial with number of trials $n$ and probability of a success $p_1$

Then:

$E(X_1|X_2>0)= \sum_{x_1=0}^n x_1 P(X_1=x_1|X_2>0)$

and we have enough information above to evaluate this.

CB

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