Paper 2, Section II, F

Probability | Part IA, 2016

Let XX be a non-negative random variable such that E[X2]>0\mathbb{E}\left[X^{2}\right]>0 is finite, and let θ[0,1]\theta \in[0,1].

(a) Show that

E[XI[{X>θE[X]}]](1θ)E[X]\mathbb{E}[X \mathbb{I}[\{X>\theta \mathbb{E}[X]\}]] \geqslant(1-\theta) \mathbb{E}[X]

(b) Let Y1Y_{1} and Y2Y_{2} be random variables such that E[Y12]\mathbb{E}\left[Y_{1}^{2}\right] and E[Y22]\mathbb{E}\left[Y_{2}^{2}\right] are finite. State and prove the Cauchy-Schwarz inequality for these two variables.

(c) Show that

P(X>θE[X])(1θ)2E[X]2E[X2]\mathbb{P}(X>\theta \mathbb{E}[X]) \geqslant(1-\theta)^{2} \frac{\mathbb{E}[X]^{2}}{\mathbb{E}\left[X^{2}\right]}

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