Let X1 Xn be independent random variables which are U11 and

Let X1, ..., Xn be independent random variables which are U(-1,1) and define

Use Chebyshev\'s inequality to estimate P( |X_n | >= 0.05)

How large must n be for the bound to be less than or equal to 0.05?

Solution

here Xi~U(-1,1) i=1(1)n independently.

so E[Xi]=0 i=1,2,....,n

V[Xi]=(1+1)²/12=1/3          i=1,2,...........,n

so E[Xn]=n*0/n=0

V[Xn]=(V[X1]+V[X2]+.....+V[Xn])/n²=1/3n

Now, by Chebyshev\'s inequality P[|Xn-E[Xn]|>=0.05*sqrt(3n)*1/sqrt(3n)]<=1/{0.05*sqrt(3n)}²

or, P[|Xn|>=0.05]<=1/{0.05*sqrt(3n)}²   [answer]

now given that 1/{0.05*sqrt(3n)}²=0.05 or,1/(0.0025*3n)=0.05 or, 0.0025*3n=1/0.05=20 or, 3n=20/0.0025=8000 or, n=8000/3=2666.667~2667 [answer]