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Math Help - expected value of particle position

  1. #1
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    expected value of particle position

    Suppose that a particle starts at the origin of the real line and moves along the line in jumps of one unit. For each jump, the probability is p \ (0\leq p \leq1) that the particle will jump one unit to the left and the probability is 1-p that the particle will jump one unit to the right. Find the expected value of the position of the particle after n jumps.
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  2. #2
    MHF Contributor harish21's Avatar
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    Re: expected value of particle position

    suppose Y_i = 1 if the particle jumps to the right, and Y_i = -1 if it jumps to the left..
    so P(Y_i =  1) = (1-p)\;and\; P(Y_i\;=\;-1)=p
    If X_n is the position of the particle after n jumps, you have:
    X_n = \sum_{i=1}^n Y_i

    now find E(Y_i) \;and\; then\; E(X_n)
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  3. #3
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    Re: expected value of particle position

    E(Y_i)=(1)(1-p)+(-1)(p)=1-2p
    E(X_n)=\sum\limits_{i=1}^n E(Y_i)=\sum\limits_{i=1}^n 1-2p=n(1-2p)=n-2np
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