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?


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

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

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

[math] \begin{cases} W_H \text{ if } ~~ Y_1 = 1 \\ W_T \text{ if } ~~ Y_1 = 0 \end{cases} [/math]

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

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

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

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


TS Contributor
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

[math] W \times W_H^{Y_1} \times W_T^{1 - Y_1} [/math]

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


TS Contributor
Yes you are correct.

The first thing in part b) is to determine the distribution of [math] X [/math]

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

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