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    Limiting Distribution question




    Let X1,......,Xn ~ Uniform (0,1). Let Yn = X bar n ^2. Find the limiting distribution of Yn.

    We know that whenever a random variable has a limiting distribution, we can use the limiting distribution as an approximation to the exact distribution function. Is it possible to apply the Weak Law of Large Numbers and assume the 0 and 1 represent a finite mean meu (0) and finite variance sigma squared(1)? If so, state X bar n ^2 converges to meu in probability and in distribution which gives us:

    F meu of x = {0 where x < meu
    1 where x>= meu

    Any assistance would be greatly appreciated. Thanks.

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    Re: Limiting Distribution question

    Quote Originally Posted by greg6363 View Post
    Let X1,......,Xn ~ Uniform (0,1). Let Yn = X bar n ^2. Find the limiting distribution of Yn.

    We know that whenever a random variable has a limiting distribution, we can use the limiting distribution as an approximation to the exact distribution function. Is it possible to apply the Weak Law of Large Numbers and assume the 0 and 1 represent a finite mean meu (0) and finite variance sigma squared(1)? If so, state X bar n ^2 converges to meu in probability and in distribution which gives us:

    F meu of x = {0 where x < meu
    1 where x>= meu

    Any assistance would be greatly appreciated. Thanks.
    Well, no, you do not need the use of the Weak Law of Large Numbers to address your interrogative. In short, you know what the paramerters are for the regular (0, 1) uniform distribution. Hint: After all, Yn is just a monotonic function of X (or X-bar) - is it not?

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    Re: Limiting Distribution question


    A monotonic function is one that is either non-increasing or non-decreasing on the interval. So the limiting distribution of Yn would be the same as the uniform distribution?

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