Paper 2, Section II, F

(i) Define the distribution function $F$ of a random variable $X$, and also its density function $f$ assuming $F$ is differentiable. Show that

$f(x)=-\frac{d}{d x} P(X>x)$

(ii) Let $U, V$ be independent random variables each with the uniform distribution on $[0,1]$. Show that

$P\left(V^{2}>U>x\right)=\frac{1}{3}-x+\frac{2}{3} x^{3 / 2}, \quad x \in(0,1)$

What is the probability that the random quadratic equation $x^{2}+2 V x+U=0$ has real roots?

Given that the two roots $R_{1}, R_{2}$ of the above quadratic are real, what is the probability that both $\left|R_{1}\right| \leqslant 1$ and $\left|R_{2}\right| \leqslant 1 ?$

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