Conditional expectation from two independent random variables

Question

This question is a practice one from "Basic Probability Theory" (Robert B. Ash).

Let $R_1$ and $R_2$ be independent random variables, each with density $f(x)=1/2e^{-x}$, $x\ge 0$; $f(x)=1/2$, $-1\le x\le 0$; $f(x)=0$, $x<-1$. Let $R_3=R_1^2+R_2^2$. Find $E(R_3|R_1=x)$.

At first from my basic knowledge I tried to get $h(z|x) = \frac{f_{13}(x,z)}{f_1(x)}$ to calculate the conditional expectation. So to get $f_{13}(x,z)$, I tried getting a distribution function $P\{R_3\le z,R_1=x\}=F_{13}(x,z)$ and differentiate it only to realize that this can't be a right way to solve this problem.

1. Is it okay to get a joint density function from a joint distribution function? I haven't seen any example or practice, at least in this book, of an example differentiating joint distribution function to get a joint density.
2. Is joint density function approach even a right way to approach for this one? I have done an example deriving joint density of two random variables on the way to get their joint distribution function. I guess this could be just that example.
3. Any help on solving this problem?

Answer 1

You seem to be confused on a couple of points so I will try to address them in order the order you asked;

1. "Is it ok to get a joint density from a joint distribution?" Absolutely! Lets say $R_1$ and $R_2$ have joint distribution $F_{R_1,R_2}(x,y)$, then it is usually safe to consider the joint density function to be $$ f_{R_1,R_2}(x,y) = \begin{cases} \frac{\partial^2}{\partial x \partial y}F_{R_1,R_2}(x,y) & \mbox{if this derivative exists at } (x,y), \\ 0 &\mbox{otherwise}. \end{cases} $$
2. Short answer - getting the joint distribution and then conditioning would not be a good way to go here! Your idea has been to construct the complete the density of $R_3$, this is far more information than you need! The question asks only for the expectation of this random variable - this is a much simpler problem than constructing the density so you were right to think you were going the wrong way at this point.
3. As mentioned just calculating the expectation is a much simpler problem because as I am sure you have seen it behaves as a linear operator and so you have $$ \begin{align} \mathbb{E} [ R_3 | R_1 ] &= \mathbb{E} [R_1^2 + R_2^2 \; | \; R_1] \\ &= \mathbb{E}[R_1^2|R_1] + \mathbb{E}[R_2^2|R_1] \\ &= R_1^2+\mathbb{E}[R_2^2] \end{align} $$ where we have used that if $X$ is a random variable and $f$ a function then $\mathbb{E}[f(X)|X] = f(X)$ and that if $X$ and $Y$ are independent then $\mathbb{E}[Y|X] = \mathbb{E}[Y]$. So all that remains for you to do is to substitute in the value $R_1$ is conditioned on and to calculate $\mathbb{E}[R_2^2]$ using the density you have been given.

Hope that helps!