# Central Limit Theorem

• Mar 6th 2011, 03:32 PM
Laydieofsorrows
Central Limit Theorem
Two questions:

1. Let X be the mean of a random sample of size 12 from the uniform distribution on the interval (0, 1). Approximate P(1/2 < X < 2/3)

2. Let X be the mean of a random sample of size 36 from an exponential distribution with mean 3. Approximate P(2.5 < X < 4)

The first one I don't understand how to get the expectation and the variance.

The second one I have no idea because it's exponential.

Thanks for the help.
• Mar 6th 2011, 03:55 PM
pickslides
Lets attack this one at a time.

For the unifrom distribution on (a,b) in your case (0,1)

For 1) mean = $\displaystyle\frac{1}{2}(a+b)$ sd = $\displaystyle \sqrt{\frac{1}{12}(b-a)^2}$

What do you get?

Now what will you do for $\displaystyle P\left(\frac{1}{3}<\bar{X}<\frac{2}{3}\right)$ ?
• Mar 6th 2011, 09:38 PM
matheagle
I wonder how close to normality a sample of size 12 from a uniform would be.
The Berry-Esseen Theorem may help here.

In the second the mean equals the variance in an exponential.
And I'm wondering where that 1/3 came from.
• Mar 8th 2011, 01:20 PM
pickslides
Quote:

Originally Posted by matheagle
And I'm wondering where that 1/3 came from.

Me to! Twas having a bad day?
• Mar 8th 2011, 05:05 PM
matheagle
na, just a typo
but I looked around to figure out what I was missing here