Paper 2, Section II, F

Probability | Part IA, 2012

(i) Define the distribution function FF of a random variable XX, and also its density function ff assuming FF is differentiable. Show that

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

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

P(V2>U>x)=13x+23x3/2,x(0,1)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 x2+2Vx+U=0x^{2}+2 V x+U=0 has real roots?

Given that the two roots R1,R2R_{1}, R_{2} of the above quadratic are real, what is the probability that both R11\left|R_{1}\right| \leqslant 1 and R21?\left|R_{2}\right| \leqslant 1 ?

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