Question

Let X1,X2,…,Xn iid∼Unif(0,b) be n i.i.d. uniform random variables on the interval [0,b] for some positive b. Suppose n is small (i.e. n<30) so that the central limit theorem is not justified.

Find an upper bound on the probability that the sample mean is “far away" from the expectation (the true mean) of X. More specifically, find the respective upper bounds given by the Chebyshev and Hoeffding inequalities on the following probability:

P(≥|Xn−E[X]|≥ c σ /√n)where σ2=VarXi

for c=2 and c=6. Each answer is numerical.

Answer 1