ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 1 of 165

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that the square of the one sample t-statistic has the F-distribution. What are its degrees of freedom?

Explanation

The t-statistic is defined as the ratio of a standard normal variable X~N (0, 1) and the square root of Equation where Y~ Equation and n is the degree of freedom.

Equation

Then we show the square of t-statistic follows F-distribution.

Equation

We known that X is standard normal distribution, then X 2 follows a chi-square distribution with degree of freedom is 1. Let assume Z= Equation

The joint p. d. f. of Y and Z is

Equation

Equation

The range of Y and Z is greater than 0

Assume Equation

The Jacobian transformation

Equation

Then the range of Z > 0 and f > 0

The p. d. f. of f and y is

Equation

Equation

Integrating it with respect to y, we get

Equation

Equation

This integral is a gamma distribution by multiple and dividing some constant.

Equation

Equation

This is a f distribution with degree of freedom is 1 and n.

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