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# Statistical Inference Problem Set

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(i) Let {Xi} be a sequence of rv's that converges in probability to a constant a. Assume that P(Xi > 0) = 1.
a) Show that the sequence {Yi} defined by Yi = sqrt(Xi) converges in probability to sqrt(a).
b) Show that, if a>0, the sequence {Zi} defined by Zi = a/Xi converges in probability to one.

(ii) Given that N = n the conditional distribution of Y is X_2n^w. The unconditional distribution of N is Poisson (theta).
a) Calculate E(Y) and Var(Y), i.e., the unconditional moments.
b) Show that, as theta -> infinity
[Y - E(Y)]/[sqrt(var(Y)] ---d--> N(0,1)

(iii) Let X be a rv with a Student's t distribution with p degrees of freedom.
a) Calculate E(X) and Var(X).
b) Show that X^2 has an F1,p distribution, ie. an F distribution with 1 and p degrees of freedom.
c) Use the results of parts (a) and (b) to argue that, as p --> infinity, X^2 converges in distribution.

##### Solution Summary

This solution helps with a statistical inference problem set.

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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