# Joint Probabilities

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Let Z be a standard normal random variable and let V have a chi-square distribution with n-degrees of freedom. Assume that Z and V are independent and let

T = Z / √ (V/n)

Find the density of T (The distribution of T is known as the t-distribution with n degrees of freedom.)

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##### Solution Summary

Let Z be a standard normal random variable and let V have a chi-square distribution with n-degrees of freedom. Assume that Z and V are independent and let

Find the density of T (The distribution of T is known as the t-distribution with n degrees of freedom.)

##### Solution Preview

Please refer to the attachment.

A statistical distribution published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under a pseudonym, so he chose "Student." Given n independent measurements , let

(1)

where is the population mean, is the sample mean, and s is the estimator for population standard deviation (i.e., s=SQRT(V)) defined by (2)

Student's t-distribution is defined as the distribution of the random variable t which is (very loosely) the "best" ...

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