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Chapter 8: Problem 18

Show that the square of a noncentral \(T\) random variable is a noncentral \(F\) random variable.

Short Answer

Expert verified
The square of a noncentral 'T'-distributed random variable is a noncentral 'F'-distributed random variable. This fact comes from the transformation of the 'T' distribution when squared which follows the distribution of 'F'. The noncentrality parameter of the resulting 'F' distribution is the square of that of the original 'T' distribution.

Step by step solution

01

Understanding 'T' and 'F' Distributions

A noncentral 'T' distribution has a noncentrality parameter which measures the degree of noncentrality. An 'F' distribution is a ratio of two independent chi-square random variables divided by their respective degrees of freedom.
02

Squared 'T' Distribution as 'F' Distribution

If \(X\) is a noncentral Student's 't'-distributed random variable with \(v\) degrees of freedom and noncentrality parameter \(未\) , then \(X^2\) has a noncentral 'F'-distribution with \(1\) and \(v\) degrees of freedom, with noncentrality parameter \(未^2\). This is because a noncentral 't'-distribution is found as \(X = (Z + 未) / sqrt{V/v}\), where \(Z\) has standard normal distribution, \(V\) has chi-squared distribution with \(v\) degrees of freedom, and \(Z\) and \(V\) are independent. If you square this, you remove the denominator and subsequently the ratio nature of 't'-distribution turning it into an 'F' distribution.
03

Final Connection

By squaring noncentral 'T', we convert a 'T' distributed variable into a noncentral 'F' distributed variable. The noncentrality parameter of the 'F' distribution will be the square of that of the 'T' distribution.

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Most popular questions from this chapter

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