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Thread: Binominal Distribution (urgent)

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    Question Binominal Distribution (urgent)




    kindly help (or provide hints) on below question:

    Suppose that you pay $1024 to enter a coin tossing game. A biased coin with head
    probability 0.3 is tossed 10 times independently. The amount will be doubled every time a
    head is obtained and halved every time a tail appears. Denote X as the number of heads
    obtained and Y as the amount of money you end up with.
    (a) Express Y in terms of X.
    (b) Determine the expected value of Y. Is this a fair game?

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    Re: Binominal Distribution (urgent)

    If you still stuck in part a), you may think of the following way first:

    Let Y_i \sim \text{Bernoulli}(0.3) be the indicator of the i-th toss having a head such that X = \sum_{i=1}^{10} Y_i.

    Can you express the outcome after the 1st toss? If you are not sure, write it down with something like

    \begin{cases} W_H \text{ if } ~~ Y_1 = 1 \\ W_T \text{ if } ~~ Y_1 = 0 \end{cases}

    in terms of the initial wealth W. (assume the question have not given the number 1024 yet).

    Note that while Y_i is the indicator for head, we can also use 1 - Y_i as the indicator for tail. Two common way to combine the above written cases

    1. W_HY_i + W_T(1 - Y_i) in an additive model
    2. W_H^{Y_i}W_T^{1-Y_i} in a multiplicative model

    You should be familiar with the second one if you have learned something about the Binomial.

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    Re: Binominal Distribution (urgent)

    thank for the explanation. but still dont know how to incorporate "The amount will be doubled every time a head is obtained and halved every time a tail appears" in part a). could you elaborate a bit more. Thanks!

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    Re: Binominal Distribution (urgent)

    Actually I have not think of a good idea to give a good hint here as those hints I have thought of will lead to the direct final answer.

    Anyway, here it goes: Just following my first post, do you see why the wealth after the first toss is in the form of

    W \times W_H^{Y_1} \times W_T^{1 - Y_1}

    Please fill in the appropriate values for the factors W_H, W_T and try to generalize the result.

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    Re: Binominal Distribution (urgent)

    Thank for your guidance. the ans should be Y=(1024)(2)^x(.5)^(10-x), right? could you proivde hint on b) also? Thank again

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    Re: Binominal Distribution (urgent)


    Yes you are correct.

    The first thing in part b) is to determine the distribution of X

    Calculating the expected value related to summing the Binomial series (Binomial Theorem).

    i.e. You need to know how to expand (a + b)^n

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