Let X be a continuous random variable. We know that it takes values between 0 and 6 , but we do not know its distribution or its mean and variance, although we know that its variance is at most 4 . We are interested in estimating the mean of X , which we denote by h . To estimate h , we take n i.i.d. samples X1,…,Xn , which all have the same distribution as X , and compute the sample mean

Question

H=1/n*∑i=1nXi.

1. Express your answers for this part in terms of h and n using standard notation.

E[H]=


Given the available information, the smallest upper bound for Var(H) that we can assert/guarantee is:

2. Var(H)≤


Calculate the smallest possible value of n such that the standard deviation of H is guaranteed to be at most 0.01.

3. This minimum value of n is:

We would like to be at least 96% sure that our estimate is within 0.02 of the true mean h . Using the Chebyshev inequality, calculate the minimum value of n that will achieve this.

4. This minimum value of n is:

Suppose now that X is uniformly distributed on [h−3,h+3], for some unknown h . Using the Central Limit Theorem, identify the most appropriate expression for a 95% confidence interval for h . You may want to refer to the normal table.

Answer 1

1. h
2. (var)/n , so its 4/n
3. 40000
4 - not sure on cheb's inequality
5. calculate the variance of x based on uniform dist (0,6), so it equals to 3, then apply CLT resulting in H+/- 1.96 * sqrt(3)/sqrt(n)

Answer 2

4. 250,000

Answer 3

1. h
2. 4/n
3. 40000
4. 250000
5. H+/- 1.96 * sqrt(3)/sqrt(n) i.e option (c)