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.
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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