Let $X$ and $Y$ be two independent continuous random variables each uniformly distributed between $[0,1]$ (i.e., $f_x (a)=f_y (a)=1,0\leq a \leq 1$). Please find the density function of the sum of the random variables $X$ and $2Y$, i.e., $f_{X+2Y} (a)$.

Am I doing this right? :[ As you can see, I get a little confused at the bottom, and I'm just kind of stuck.

Any help would be greatly appreciated, thank you.

My work. Cumulative distribution function of $X+ 2Y$: $$ \begin{align*} F_{X+2Y} (a) &= P(X+2Y \leq a) \\ &= \iint_{x+ y \leq a} f_X(x) f_Y(y) ~ dy ~dx \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{a - 2y} f_X(x) f_Y(y) ~ dx ~dy \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{a - 2y} f_X(x) ~ dx\ f_Y(y) ~dy \\ &= \int_{-\infty}^{\infty} F_X(a-2y) f_Y(y) ~dy \end{align*} $$

The probability density function: $$ \begin{align*} f_{X+2Y}(a) &= \frac{d}{da} \int_{-\infty}^{+\infty} F_X(a-2y) f_Y(y) ~ dy \\ &= \int_{-\infty}^{+\infty} \frac{d}{da} F_X(a-2y) f_Y(y) ~ dy \\ &= \int_{-\infty}^{+\infty} f_X(a-2y) f_Y(y) ~ dy \end{align*} $$

For $[0,1]$: $$ f_X(a) = f_Y(a) = \begin{cases} 1, & 0 \lt a \lt 1, \\ 0, &\text{otherwise}. \end{cases} $$

$$ f_{X + 2Y}(a) = \int_{-\infty}^{+\infty} f_X(a-2y) f_Y(y) ~ dy \quad \text{for } 0 \lt a \lt 1. $$

$$ f_{X + 2Y}(a-2) = \int_0^a dy = a. $$

  • $\begingroup$ You have the right ideas but you are using $f_{X+2Y}$ to denote both the CDF and the pdf: be careful. Also, if you would try and draw a diagram of the region where $f_{X,Y}(x,y)$ is nonzero and look at the regions of integration, you might want to modify the limits of various integrals appropriately. $\endgroup$ – Dilip Sarwate Nov 25 '11 at 18:26
  • $\begingroup$ You are right. My CDF should of said Fx+2y(a) not fx+2y(a). As far as the diagram goes, I'm afraid I don't understand the problem well enough to follow. I have the answer to a simliar problem X+Y that I am trying to follow, but it is sadly not helping. $\endgroup$ – Ubez Nov 25 '11 at 18:30
  • $\begingroup$ Draw a plane with coordinate axes $x$ and $y$. Draw on the plane the region where $f_{X,Y}$ is nonzero. Draw on the plane $4$ lines: $x+2y \leq -1$, $x+2y \leq 0.5$, $x+2y \leq 1.5$, $x+2y \leq 3$. THINK about the fact that a double integral will give you the volume between the plane and the $f_{X,Y}$ surface in the region over which you are integrating. $\endgroup$ – Dilip Sarwate Nov 25 '11 at 18:36

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