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Math Help - Central Limit Theorem

  1. #1
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    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.
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  2. #2
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    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) ?
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  3. #3
    MHF Contributor matheagle's Avatar
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    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.
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  4. #4
    Master Of Puppets
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    Quote Originally Posted by matheagle View Post
    And I'm wondering where that 1/3 came from.
    Me to! Twas having a bad day?
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  5. #5
    MHF Contributor matheagle's Avatar
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    na, just a typo
    but I looked around to figure out what I was missing here
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