Creating a pmf for a coin toss situation

Question

Suppose that George starts with one dollar and will gamble with it as follows: if a fair coin toss results in heads he’ll win a dollar, and if it results in tails he’ll lose a dollar; and he’ll continue to bet a dollar on subsequent flips for up to three flips, but he’ll quit as soon as he loses the first time. Letting $X$ denote George’s net profit, give the pmf of $X$ (being sure to indicate a value for every real number).

Attempted Solution

Let $H,T$ denote heads and tails, respectively. Then the sample space is

${T}$, ${HT}$, ${HHT}$, ${HHH}$

These give net profits of $-1, 0, 1$, and $3$, respectively and have probabilities of $1/2, 1/4, 1/8$, and $1/8$, respectively.

Then $$p_X(x) = \begin{cases} {\frac{1}{8}} & \text{$x = 3$} \\ \frac{1}{8} & \text{$x=1$} \\ \frac{1}{4} & \text{$x=0$} \\{\frac{1}{2}} & \text{$x = -1$}\\{0} & \text{otherwise}\end{cases}$$

Did I do this correctly?

Answer 1

Did I do this correctly?

No. Well, almost.

You have the correct probability masses for each outcome in the sample space.   However, the intervals in your piecewise function donut make any sense in light of the fact that the support should be $\{-1,0,1,3\}$, and so the probability mass function will have value zero at any other value of your discrete random variable.