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Math Help - Expected Value of a "composite" Random Variable

  1. #1
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    Expected Value of a "composite" Random Variable

    Hi everybody! I'm presented with the following problem:
    Consider a mouse (or rabbit, or other dumb aninal of your choice) that is inside a room (or cell) with THREE doors. The doors behave as follows:

    • The first door leads to a tunnel that will take the mouse back to the room after two days
    • The second door leads to a tunnel that will take it back to the room after four days
    • The third door will lead the mouse to freedom after one day

    Other assumptions:

    • Every time the mouse is in the room, it chooses one door randomly, with equal probability.
    • The time the mouse spends choosing a door can be disconsidered
    • The mouse is dumb enough to choose the same door more than one time. i.e., it can theoretically stay in the room forever if it never chooses the right door.

    Given this situation, we ask, how long does it probably take until the mouse REACHES freedom?

    Now my thought process: I believe I have to define a random varibale T that represents the time the mouse stays not in freedom. And then the expected value of T is the answer to the problem.

    I believe I have the correct 'outline' to solve the problem, but I'm struggling with the individual steps. In particular, I couldn't manage to define the T variable well, and then find its PDF to calculate the expected value.

    So, any tips on how to start? (sorry for the verbose post!)
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  2. #2
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    Will you be able to decipher the following non-verbose computation of the expected value?

    E[T]=0\cdot\frac{1}{3}+(2+E[T])\frac{1}{3}+(4+E[T])\frac{1}{3}

    hence E[T]=6.

    For the definition of T, you should start by defining an (infinite) sequence of r.v.s corresponding to the successes choices of a door, then define the index N where the sequence stops and finally T using the previous r.v.s. (Then you can justify rigorously the above computation of the expectation)
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  3. #3
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    So let's check if I understand:

    I define the RV D as the choice of a door and then the sequence \{D_i\} where each D_i \in \{1,2,3\} with probability \frac{1}{3} for each value.

    Then I define the index where the sequence 'stops', which I think is also a RV (but I don't know how its values would be distributed! shouldn't the values depend on the amount of doors? i.e., with more doors, it would be more likely that the index is greater? or not?!)

    So I think I still can't define T rigorously. Any more tips?

    Anyway, I think I 'get' the intuition behind the expected value calculation, but still couldn't figure out how to justify it formally.
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  4. #4
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    So you have N=\inf\{i\geq 1|D_i=3\}, and if you remark that after opening door number d\in\{1,2\} the mouse has to wait 2d days, then it should be clear that T=2D_1+2D_2+\cdots+2D_{N-1}. This is a fine definition for T. By the way, when defining (D_i)_{i\geq 1}, you should specify that these random variables are independent.

    In order to justify the computation of E[T], you should start by splitting the expectation according to the value of D_1, and use the fact that, given D_1\neq 3, the "remaining time" \widetilde{T}=T-2D_1 has same distribution as T (and therefore same expectation).
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