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

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

### 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 where Y~ and n is the degree of freedom.

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

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=

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

The range of Y and Z is greater than 0

Assume

The Jacobian transformation

Then the range of Z > 0 and f > 0

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

Integrating it with respect to y, we get

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

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