. Suppose that X has the t distribution with m degrees of

Chapter 8, Problem 1

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QUESTION:

. Suppose that X has the t distribution with m degrees of freedom (m > 2). Show that Var(X) = m/(m 2). Hint: To evaluate E(X2), restrict the integral to the positive half of the real line and change the variable from x to y = x2 m 1 + x2 m . Compare the integral with the p.d.f. of a beta distribution. Alternatively, use Exercise 21 in Sec. 5.7

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QUESTION:

. Suppose that X has the t distribution with m degrees of freedom (m > 2). Show that Var(X) = m/(m 2). Hint: To evaluate E(X2), restrict the integral to the positive half of the real line and change the variable from x to y = x2 m 1 + x2 m . Compare the integral with the p.d.f. of a beta distribution. Alternatively, use Exercise 21 in Sec. 5.7

ANSWER:

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Let Y and Z be two independent random variables from  distribution with  degrees of freedom and from the standard normal distribution, respectively. Random variable defined as

is said to have  distribution with  degrees of freedom.

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