Question

Let (X1,…,Xn)∼Unif(0,b),b>0
. Find E[∑XiX(n)]

Answer 1

To find E[∑XiX(n)], we use the Central Limit Theorem to approximate the mean of the random variable ΣXiX(n) as (n)(µx), where µx is the mean of X.

Step-by-step explanation:

To find E[∑XiX(n)], we first need to determine the distribution of the random variable ΣXiX(n). Since Xi∼Unif(0,b) for all i, we can consider X1, X2, ..., Xn as n independent and identically distributed random variables.

Using the Central Limit Theorem, as n increases, the random variable ΣXiX(n) tends to be normally distributed. Therefore, we can approximate the mean of ΣXiX(n) as (n)(µx), where µx is the mean of X.

So, E[∑XiX(n)] ≈ (n)(µx). Since X∼Unif(0,b), the mean of X, µx, can be calculated as µx = (a+b)/2 = (0+b)/2 = b/2.