problem 50

5.2.1 problem 50

By symmetry, and by X and Y being independent and identically distributed, the chance that X should be smaller than Y should not be different than the chance that Y should be smaller than X. Furthermore, from independence, joint pdf, $g (x, y) = f_{X} (x) f_{Y} (y)$

\begin{matrix} P (X < Y) & = \int_{y = - \infty}^{\infty} \int_{t = - \infty}^{y} f_{X} (x) f_{Y} (y) dtdy & (5.12) = \int_{y = - \infty}^{\infty} F (y) f (y) dy & (5.13) = \frac{1}{2} & (5.14) \end{matrix}

When X and Y are not independent say X = Y + 1, (assume the existence of such X and Y), then, $P (X < Y) = 0$ and $P (Y < X) = 1$ When X and Y are not identically distributed, say $X \sim Unif (0, 1)$ and $Y \sim Unif (- 1, 0)$ then, $P (X < Y) = 1$ and $P (Y < X) = 0$

[next] [front] [up]