From the given image, it appears to be a probability problem involving a probability mass function $f(x)$ and transformations of a random variable $Y$. I will solve it step-by-step. Let’s address the task:

Given:

1. ( f(x) = \begin{cases} p(1-p)^{x-1}, & x = 1, 2, 3, \dots \ 0, & \text{otherwise} \end{cases} )

2. Different transformations for $Y$ based on $X$:

   • (a) $Y = \frac{1}{x+1}$
   • (b) $Y = x^2$
   • (c) ( Y = \begin{cases} (2 - |x - 3|), & x \leq 5 \ (x - 5), & x > 5 \end{cases} )

To Determine:

The probability mass function (PMF) of $Y$.

Let’s compute these:

(a) $Y = \frac{1}{x+1}$

Here, $Y$ is a function of $X$. To derive the PMF of $Y$, we find the relationship between $X$ and $Y$. Specifically: $Y = \frac{1}{x+1} \implies x = \frac{1}{Y} - 1$ The PMF of $Y$ will depend on substituting $x$ into $f(x)$: $P(Y = y) = f(x), \text{ where } x = \frac{1}{y} - 1$ Thus: $P(Y = y) = p(1-p)^{\frac{1}{y} - 2}, \text{ for } y = \frac{1}{2}, \frac{1}{3}, \dots$

(b) $Y = x^2$

Here, $Y$ is a quadratic function of $X$: $Y = x^2$ Possible values of $Y$ are $1, 4, 9, \dots$ (squares of positive integers). For each $y$: $P(Y = y) = P(X = \sqrt{y})$ If $\sqrt{y}$ is an integer, then: $P(Y = y) = f(\sqrt{y}) = p(1-p)^{\sqrt{y} - 1}$ Otherwise, $P(Y = y) = 0$.

**(c) ( Y =

 \begin{cases} 
 (2 - |x - 3|), & x \leq 5 \\
 (x - 5), & x > 5 
 \end{cases} \)**

For $x \leq 5$: $Y = 2 - |x - 3|$ Possible values of $Y$ depend on the range of $x$. Compute $Y$ for $x = 1, 2, 3, 4, 5$:

• $x = 1$: $Y = 0$
• $x = 2$: $Y = 1$
• $x = 3$: $Y = 2$
• $x = 4$: $Y = 1$
• $x = 5$: $Y = 0$

Thus, the PMF for $Y$ for $x \leq 5$: $P(Y = 0) = f(1) + f(5), \quad P(Y = 1) = f(2) + f(4), \quad P(Y = 2) = f(3)$

For $x > 5$: $Y = x - 5$ Here, $Y$ takes values $1, 2, 3, \dots$ (for $x = 6, 7, 8, \dots$): $P(Y = y) = f(y + 5)$

Let me know if you want further details, have any questions, or need these results expanded further.

Related Questions:

1. What is the cumulative distribution function (CDF) of $Y$ for each case?
2. How does the PMF change if $f(x)$ is redefined for a finite $x$ range?
3. What is the expected value $E[Y]$ for each transformation?
4. How would you compute the variance of $Y$ in case (c)?
5. How do these results generalize for other transformations of $X$?

Tip:

Always carefully identify the relationship between the original random variable $X$ and the transformed random variable $Y$ when working with PMFs.