What Is the Distribution of 2(X̄ - 10) / S?

[buddingscientist]

1. If X1, X2, X3, X4 are a random samle from a normal distribution with mean 17, then what is the distribution of 2(X - 10) / S 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 P ( (X / Cn) \leq x) = P (Z \leq x), Z ~ N(0,1) ?


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


4. What is the distribution of X/Y where X and Y are independant {X^{2}}_{1} 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) = e^{1/n} -1.

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


Thanks for your time

 

[TenaliRaman]

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

 

[BicycleTree]

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.