Solution to Problem 2.

(2) [15 points] Let $X_1$ and $X_2$ be i.i.d. uniform(0,1) rv's and $Y_1 = \min\{X_1,X_2\}$, $Y_2 = \max\{X_1,X_2\}$. Show that the conditional distribution of $Y_1$ given $Y_2 = y_2$ is uniform(0,$y_2$).

Solution: It is fairly easy to show that the joint pdf of $(Y_1,Y_2)$ is

$$f(y_1,y_2) \; = \; 2 , \quad 0 < y_1 < y_2 < 1,$$

and the marginal pdf for $Y_2$ is

$$f(y_2) \; = \; 2y_2 , \quad 0 < y_2 < 1.$$

Thus, the conditional pdf for $Y_1$ given $Y_2 = y_2$ is

$$f(y_1\vert y_2) \; = \; \frac{2}{2 y_2} \; = \; 1/y_2 , \quad 0 < y_1 < y_2 < 1.$$

As a function of $y_1$, we recognize this as the pdf for the uniform($y_2$) distribution.