Help me with this question. This is the question 5.7 of the book Hoel
Let (u, v) be chosen uniformly in the square 0<=u<=1 and 0<=v<=1. Let x be the random variable that associate the number u+v to the point (u, v). Get the Distribution function of x.
1 Answer
Let $f_X$ be the full density function of$X$, and let $f_Y$ be the full density function of $Y$. Let $Z=X+Y$. Then the pdf of $Z$ is given by
$$f_Z(z)=\int_{-\infty}^\infty f_X(x)f_Y(z-x)\,dx. (*)$$
Note that $f_x =1$ for $x\in(0,1)$ and $f_y =1$ for $y\in(0,1)$. Using (*) we can easily see that $f_Z(z)=0$ for $z\lt 0$, and also for $z\ge 2$. Next we compute $f_Z$ on $(0,2)$. We break it down into two cases (i) $0\lt z\le 1$ and (ii) $1\lt z\lt 2$.
(i) The product $f_X(x)f_Y(z-x)$ is ither $0$ or $1$. We have $f_Y(z-x)=1$, if $z-x\ge 0$, that is, $x\le z$. Hence in this case
$$f_Z(z)= \int_0^z 1\,dx=z, 0\lt z\le 1.$$
(ii) Suppose that $1\lt z\lt 2$. Then $f_Y(z-x)$ to be $1$, if we have $z-x\le 1$, that is $x\ge z-1$. Thus
$$f_Z(z)=\int_{z-1}^1 1\,dx=2-z, 1\lt z\lt 2.$$

-
If you like the solution, a tip is appreciated.
- 1 Answer
- 282 views
- Pro Bono
Related Questions
- Central Limit Theorem question
- Random Walk on Cube
- Fisher Information and Wilson Confidence Interval. B+D only.
- What are the odds of drawing the two tarot cards that I wanted, then reshuffling the pack, and drawing them both again straight away, in the same order?
- Density plot
- Let $X$ be a single observation from the density $f(x) = (2θx + 1 − θ)I[0,1](x)$ with $−1≤ θ ≤ 1$. Find the most powerful test of size $α$ and its power
- Question about COEFFICIENT OF CORRELATION
- What Are The Odds ?