1. ## 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?

2. ## Re: Binominal Distribution (urgent)

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

Let be the indicator of the -th toss having a head such that .

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

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

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

2. in a multiplicative model

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

3. ## 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!

4. ## 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

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

5. ## 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

6. ## Re: Binominal Distribution (urgent)

Yes you are correct.

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

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

i.e. You need to know how to expand

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