Paper 2, Section I, F

Probability | Part IA, 2013

Let XX be a random variable with mean μ\mu and variance σ2\sigma^{2}. Let


Show that G(a)σ2G(a) \geqslant \sigma^{2} for all aa. For what value of aa is there equality?



Supposing that XX has probability density function ff, express H(a)H(a) in terms of ff. Show that HH is minimised when aa is such that af(x)dx=1/2\int_{-\infty}^{a} f(x) d x=1 / 2.

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