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Statistical 'Inference'

  1. May 19, 2005 #1
    1. If X1, X2, X3, X4 are a random samle from a normal distribution with mean 17, then what is the distribution of [itex]2(X - 10) / S[/itex] where X should be X bar.

    Our notes are just awful for this topic.
    Any tips how to proceed with this one, and what is S?


    2. Let X1, X2, ... be a sequence of independant and identically distributed random variables with mean zero and variance 36. What is the sequece Cn for which

    lim p -> inf [itex] P ( (X / Cn) \leq x) = P (Z \leq x), Z ~ N(0,1) ?[/itex]


    Would this just be sqrt(Var) = sqrt(36) = 6 ?


    4. What is the distribution of X/Y where X and Y are independant [itex]{X^{2}}_{1}[/itex] random variables? (chi-squared with 1 deg. of freedom)

    Not sure how to progress with this one, do we play with the degrees of freedom to get some sort of trivial answer?



    5. Is theta-hat consistent if MSE(theta-hat) = [itex]e^{1/n} -1[/itex].

    Since MSE approaches 0 as n approaches infinity (n-> inf, MSE -> [itex]e^{0} -1[/itex] -> 0) then yes, theta-hat is consistent. ?


    Thanks for your time
     
  2. jcsd
  3. May 23, 2005 #2
    1. S is most likely to be standard deviation
    2. I have no idea what this question is about
    3.
    There is a theorem which states that
    "if X1 and X2 are two indepdendent chi-squared variates with n1 and n2 d.f resp., then X1/X2 is a Beta2(n1/2,n2/2)"
    Direct use of this theorem makes this one easy. I am not sure if u are aware of this theorem. If u want i can prove this one.

    -- AI
     
  4. May 23, 2005 #3
    I think you need to know the standard deviation of X before you can answer 1. S is probably the sample standard deviation, defined as the sum over all i of (Xi - Xbar)/(n - 1).

    Your notation on 2 is confusing. I believe you meant: lim p --> inf (P(Xp/Cp < x) = P(Z < x)) Assuming you meant that, it's probably intended that X is distributed normally. If it is distributed normally, then all you need to do is know how to transform a normal distribution with mean 0 into a standard normal distribution.

    For 4, you can also use the F distribution. If X1 and X2 are 2 independent RV's distributed as chi-squares with a and b degrees of freedom respectively, then (X1 / a) / (X2 / b) has the F distribution with a and b degrees of freedom.

    I can't help you with 5.
     
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