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

Question

1) 2 X1-3μ

2) (X1-μ)/2 σ

3) 2Xn -4

Thankyou

Answer 1

X has expectation E(x)=mu ,variance E((X-mu)^2) = sigma.

Looking at (1)  2X-3mu,

its expectation is 2mu - 3 mu , i.e -mu

and its variance = E( 2X-3mu)^2)

= E(2X^2) - 6mu E(x) + 9mu^2

= 2E(X^2) - 4 E(X) mu + 2mu^2  -2muE(X) + 7mu^2

=   sigma^2                            - 2 mu^2    + 7 mu^2

You can repeat the method for (2)  and (3).

Answer 2

Answer one is incorrect.

Consider this: once you multiply the distribution by a constant (e.g., by 2 in problem 1), any point (e.g., 0) must be the same number of standard deviations away from the new mean as it was before you multiplied it.  Thus (μ/σ) = 2(μ/σ) =2μ/(2σ).

Thus, the standard deviation for 2μ is 2σ.  The variance is thus (2σ)^2, or 4σ^2.  The solution for the mean of the random sample from answer one is correct.  Thus, the distribution of 2X - 3μ ~ N(-μ, 4σ^2).

The same process can be repeated for #2 and #3.

As further support that this answer is correct, and answer 1 is incorrect, see the wiki entry for normal distribution here: http://en.wikipedia.org/wiki/Normal_dist...

Under the "Properties" tag, see where bullet point one.  That clearly states the principle in mathematical terms.

Good luck with your homework!