Question

Equality in Chebychev's inequality. Let and k be three numbers, with ơ > 0 and k 2 1. Let X be a random variable with the following distribution 2k2 if otherwise. k 2 0 a) Sketch the histogram of this distribution for μ-0, σ 10, k-1, 2, 3. b) Show that E(X-μ, Var(X) σ2. P(X-1-kr)-1/k2. So there is equality in Chebychev's inequality for this distribution of X. This means Chebychev's inequality cannot be improved without additional hypotheses on the dis- tribution of X c) Show that if Y has E(Y) = μ, Var(Y) = σ2 , and PlY-μ| 〈 σ) = 0, then Y has the same distribution as X described above for k1

Answer 1

$(b)~E(X)=(\mu+k\sigma)\times \frac{1}{2k^2}+(\mu-k\sigma)\times \frac{1}{2k^2}+\left ( 1-\frac{1}{k^2} \right )\mu\\ =\mu\\ Var(X)=E(X-\mu)^2\\ =(\mu+k\sigma-\mu)^2\times \frac{1}{2k^2}+(\mu-k\sigma-\mu)^2\times \frac{1}{2k^2}+\left ( 1-\frac{1}{k^2} \right )(\mu-\mu)^2\\ =\sigma^2\\ P(|X-\mu|\geq k\sigma)=1-P(|X-\mu|< k\sigma)\\ =1-P(\mu-k\sigma<X<\mu+k\sigma)=1-P(X=\mu)=1-\left ( 1-\frac{1}{k^2} \right )=\frac{1}{k^2}\\ Chebyshev's~inequality:\\ If~X~be~a~r.v.~such~that~E(X)=\mu,~Var(X)=\sigma^2~then~P(|X-\mu|\geq t\sigma)\leq \frac{1}{t^2}\\ for~any~t>0.~=~holds~if~X-\mu=-t,~0~and~t~with~probabilities\\ ~p,1-2p~and~p~respectively.\\ Hence~in~this~case~P(|X-\mu|\geq k\sigma)~attains~the~lower~bound.\\ c.~P(|Y-\mu|<\sigma)=0\Rightarrow P(Y=y)=0~if~y\in (\mu-\sigma,~\mu+\sigma)\\ and~P(|Y-\mu|\geq \sigma)=1.~Hence~from~Chebychev's~inequality,\\ ~P(|Y-\mu|\geq \sigma)~attains~its~lower~bound~with~k=1~and~its~distribution~same~as~X\\ ~with~k=1$