Let U and V be independent continuous random variables, identically distributed uniformly over [0,1]

Question

Let $U$ and $V$ be independent continuous random variables, identically distributed uniformly over $[0,1]$. Show that for $0 \leq x\leq1$ , $$P(x < V < U^2)= \frac{1}{3} - x + \frac{2}{3} x^{3/2}$$ and hence write down the density of $V$ conditional on the event $V< U^2$.

How should I introduce the conditional probability aspect into the solution ? Can this be rewritten as the cumulative distribution of joint variables ?

Answer 1

Assuming you've proved the first part.

Using: $P(A\mid B) = \frac{P(A\cap B)}{P(B)}$.

Then $P(x<V\mid V<U^2) = \frac{P(x<V<U^2)}{P(V<U^2)}$. Then:

$$P(V\leq x\mid V<U^2)=1-P(x<V\mid V<U^2)$$ is the CDF of $V$ given that $V<U^2$.

So the derivative of the CDF would be the PDF of $V$ given $V<U^2$.

So $P(V<U^2)=P(0<V<U^2)=\frac{1}{3}$ and $P(x<V<U^2)=\frac{1}{3}-x+\frac{2}{3}x^{3/2}$. The rest is just arithmetic.

Answer 2

Here is a hint. Draw a square in a Cartesian coordinate plane with vertices $(0,0), (1,0), (1,1), (0,1)$. Label the horizontal axis $V$, and the vertical axis $U$. Draw a vertical line $V = x$ for some $x \in [0,1]$. Draw the curve $V = U^2$, which is a parabola with vertex at $(0,0)$ and axis of symmetry coinciding with the line $U = 0$. Now consider the area of the region inside the square that satisfies $x < V < U^2$ as a function of $x$.