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Math Help - Expected values and probability mass function.

  1. #1
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    Expected values and probability mass function.

    Hi everyone. I am incredibly lost on this problem and I have no idea how to do it. It seems like it's easy, but I'm just not getting it. Any hints would be extremely helpful. Thanks!

    Let X be a random variable taking values 0, 1 or 2. Suppose E[X] = m_1
    and E[X^2] = m_2. Compute P [X = i] for i = 0; 1; 2.

    (E is expected value; P is the probability function.)
    Last edited by mr fantastic; July 16th 2011 at 02:20 PM. Reason: Title.
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    Re: incredibly lost... please help... probability mass functions

    Hi everyone. I am incredibly lost on this problem and I have no idea how to do it. It seems like it's easy, but I'm just not getting it. Any hints would be extremely helpful. Thanks!

    Let X be a random variable taking values 0, 1 or 2. Suppose E[X] = m_1
    and E[X^2] = m_2. Compute P [X = i] for i = 0; 1; 2.

    (E is expected value; P is the probability function.)
    write some simultaneous equations containing the information in the question:

    E(X) =m_1 = 0 \cdot P(X=0)  + 1 \cdot P(X=1)  + 2 \cdot P(X=2)
    E(X^2) =m_2 = 0 \cdot P(X=0)  + 1^2 \cdot P(X=1)  + 2^2 \cdot P(X=2)

    ie,

    m_1 = P(X=1)  + 2 \cdot P(X=2)
    m_2 = P(X=1)  + 4 \cdot P(X=2)


    Solve simultaneously to get P(X=1) and P(X=2). Finally, note that P(X=0) + P(X=1) + P(X=2) =1
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    Re: incredibly lost... please help... probability mass functions

    Quote Originally Posted by iamthemanyes View Post
    Let X be a random variable taking values 0, 1 or 2. Suppose E[X] = m_1 and E[X^2] = m_2. Compute P [X = i] for i = 0; 1; 2.
    To shorten the notation let p_i=\mathcal{P}(X=i),~i=0,1,2

    From the given \mathcal{E}[X]=0\cdot p_0+1\cdot p_1+2\cdot p_2=m_1
    AND
    \mathcal{E}[X^2]=0^2\cdot p_0+1^2\cdot p_1+2^2\cdot p_2=m_2.

    Now you can solve for p_1~\&~p_2.

    Then recall that p_0+p_1+p_2=1.
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