Find $ \mathbb P \{ 2X + 3Y \gt 2 \}$, given the CDFs of $X$ and $Y$.

Question

The Statement of the Problem:

Suppose that $X$ and $Y$ are independent random variables with $F_X(x) = 2e^{-2x}, x \gt 0,$ and $F_Y(y) = 3e^{-3y}, y \gt 0.$

Find $ \mathbb P \{ 2X + 3Y \gt 2 \}$.

SOLUTION:

The solution states that $2X \sim 2\text{Expo}(2) \sim \text{Expo}(1)$ and $2Y \sim 3\text{Expo}(3) \sim \text{Expo}(1)$ and proceeds from there by identifying the sum of the two with a gamma-distributed random variable. This, of course, implies that $X \sim \text{Expo}(2)$ and $Y \sim \text{Expo}(3)$. However, what's given above in the statement of the problem I take to be the CDFs of these random variables (because of the capital "$F$"s), not the PDFs. And so, differentiating the above (to obtain the PDFs), we have

$$f_X(x) = -4e^{-2x} \text{ and } f_Y(y) = -9e^{-3y} $$

which are obviously not the PDFs of two exponentially distributed random variables with parameters $2$ and $3$, respectively. Is this mistake, or is this there some "trick" I'm missing here?

Answer 1

Obviously, the given functions "$F_X$" and "$F_Y$" must be density functions. If they were CDF's, they must be non decreasing. However, both of them are decreasing. It must be a mistake!

Answer 2

For finding out the sum of two independent exponentially distribyted random variables.Just use the transformation $X=U-V$ and $Y=V$ and find the joint pdf of $U$ and $V$ $$f_{U,V}(u,v)=\begin{cases} e^{-(u-v)}. e^{- v}.1&\text{if $0<u-v<\infty\,\,\,\,$ $0<v<\infty\;\;$ $0<u<\infty$}\\ 0, & \text{otherwise} \end{cases} $$ just integrate with respect to $v$ t get the pdf of $U$ i.e $$f_U(u)=\int_{0}^ue^{-u}dv\,\,\,\,\,\,\,,0<u<\infty$$ Thus u get $$f_U(u)=ue^{- u}\,\,\,\,\,\,\,\,0<u<\infty$$ Just use this pdf of the sum of two independent exponential random variables.and just find the probability by integrating $$P(U>2)=\int_{2}^\infty f_U(u)du $$ Let me also clear the fact that what you have been given is not the Cdf its actually the pdf..you can check that by verifying the properties of cdf and pdf