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Math Help - Conditional Expectation problem

  1. #1
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    Conditional Expectation problem

    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.
    Last edited by downthesun01; August 3rd 2011 at 11:22 PM.
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  2. #2
    Grand Panjandrum
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    Re: Conditional Expectation problem

    Quote Originally Posted by downthesun01 View Post
    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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