Can any one State the distribution of this?

Question

Suppose X1…Xn is a random sample from the N(μ, σ^2) distribution. State the distribution of each of the following statistics:

i) Xn – 2 μ = 1/n ∑ (i=1 to n) Xi - 2 μ

ii) (Xn – μ)/ σ

iii) 2Xn – 4

Answer 1

Since each of these is a linear combination of normal variables, they are all normal variables. We just need to compute their means and variances.

i) I take this to be the distribution of Xbar - 2 mu. Its mean is just mu - 2 mu = - mu. Its variance is trickier.

(Xbar - 2 mu)^2 = n/n^2 * sum of all X_i ^2  + 1/n^2 * sum of all distinct X_i * X_j - 4 mu Xbar + 4 mu^2.

The expected value of this expression is

(sigma^2 + mu^2) / n + (n^2 - n)/n^2 * mu^2,

which simplifies to sigma^2 / n + mu^2.

Using var(Y) = E(Y^2) - E(Y)^2 for the above Y, we get that the variance of Xbar - 2 mu is

sigma^2 / n . This solves i).

ii) This is done similarly. Again calculate E(Y^2) - E(Y)^2 for this Y to get its variance. Again combine the equal terms and combine the unequal terms (that is, X_i * X_i and X_i * X_j). There are n equal terms and n^2-n unequal terms. This time the mean is 0 and the variance is again sigma^2 / n .

iii) Similarly the mean is 2 mu - 4 and the variance is

4(sigma^2 / n + mu^2) - 16 mu^2 + 16 which simplifies a bit.