# Two independant and identically distributed uniform distributions

• Apr 19th 2009, 10:16 AM
djones90
Two independant and identically distributed uniform distributions
Let U and V be independent and identically distributed uniformly on the interval [0, 1].

Show that for 0<x<1;

P(x<V<U^2)=1/3 - x + (2/3)*x^3

and hence write down the density of V conditional on the event that V<U^2.
• Apr 20th 2009, 05:28 AM
mr fantastic
Quote:

Originally Posted by djones90
Let U and V be independent and identically distributed uniformly on the interval [0, 1].

Show that for 0<x<1;

P(x<V<U^2)=1/3 - x + (2/3)*x^3

and hence write down the density of V conditional on the event that V<U^2.

1. Note that $f(u, v) = 1$ for $0 \leq u \leq 1$, $0 \leq v \leq 1$ and zero otherwise.

2. Draw the square defined by $u = 0, u = 1, v = 0$ and $v = 1$ and shade the region corresponding to $x < v < u^2$.

3. Then $\Pr(x < V < U^2) = \int_{u = \sqrt{x}}^{u = 1} \int_{v = x}^{v = u^2} 1 \, dv \, du$.

I get $\frac{1}{3} - x + \frac{2}{3} x^{\color{red}{3/2}}$, NOT $\frac{1}{3} - x + \frac{2}{3} x^{\color{red}{3}}$ ....
• Apr 22nd 2009, 02:49 AM
djones90
my mistake.

Thanks!