Paper 2, Section II, 12F

Probability | Part IA, 2021

State and prove Chebyshev's inequality.

Let (Xi)i1\left(X_{i}\right)_{i \geqslant 1} be a sequence of independent, identically distributed random variables such that

P(Xi=0)=p and P(Xi=1)=1p\mathbb{P}\left(X_{i}=0\right)=p \text { and } \mathbb{P}\left(X_{i}=1\right)=1-p

for some p[0,1]p \in[0,1], and let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be a continuous function.

(i) Prove that


is a polynomial function of pp, for any natural number nn.

(ii) Let δ>0\delta>0. Prove that

kKδ(nk)pk(1p)nk14nδ2\sum_{k \in K_{\delta}}\left(\begin{array}{l} n \\ k \end{array}\right) p^{k}(1-p)^{n-k} \leqslant \frac{1}{4 n \delta^{2}}

where KδK_{\delta} is the set of natural numbers 0kn0 \leqslant k \leqslant n such that k/np>δ|k / n-p|>\delta.

(iii) Show that

supp[0,1]f(p)Bn(p)0\sup _{p \in[0,1]}\left|f(p)-B_{n}(p)\right| \rightarrow 0

as nn \rightarrow \infty. [You may use without proof that, for any ϵ>0\epsilon>0, there is a δ>0\delta>0 such that f(x)f(y)ϵ|f(x)-f(y)| \leqslant \epsilon for all x,y[0,1]x, y \in[0,1] with xyδ|x-y| \leqslant \delta.]

Typos? Please submit corrections to this page on GitHub.