Question

Consider X1,X2,…,Xn, where X is an i.i.d. uniform random variable on [0,1]. What is the distribution of the sample mean Xˉ for a sample size n?

A. Uniform Distribution on [0,1]
B. Normal Distribution with mean μ=0.5 and standard deviation σ=1/√12n
C. Exponential Distribution with rate parameter λ=n
D. Chi-Squared Distribution with n degrees of freedom

Answer 1

The distribution of the sample mean for a sample size n follows a Normal Distribution with its mean remaining at 0.5 and the standard deviation being 1/√(12n), according to the central limit theorem.

Step-by-step explanation:

The distribution of the sample mean Ö for a sample size of n is a Normal Distribution with mean μ=0.5 and standard deviation σ=1/√(12n), so the correct answer is B. Normal Distribution with mean μ=0.5 and standard deviation σ=1/√12n.

The central limit theorem states that, given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the original distribution of the variable. Since the individual observations Xi are i.i.d. uniform random variables on the interval [0,1], we know the mean (μ) of this uniform distribution is 0.5 and its variance (σ2) is 1/12.

When calculating the mean of the sample Ö, which is the sum of these uniform variables divided by the sample size n, the expected value does not change but the variance of the sampling distribution decreases. Specifically, the variance of the sample mean Ö is σ2/n which equates to (1/12)/n or 1/(12n) and thus the standard deviation would be 1/√(12n).