If a random variable U has the t-distribution with n degrees of freedom, show that Z = U2 has the F-distribution with 1 and n degrees of freedom.
To show that the square of a random variable following a t-distribution with n degrees of freedom, i.e., U^2, has an F-distribution with 1 and n degrees of freedom, we can use the probability density functions (PDFs) and cumulative distribution functions (CDFs) of these distributions.
First, let's denote U ~ t(n) as the random variable following a t-distribution with n degrees of freedom.
The PDF of the t-distribution with n degrees of freedom is
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Prove that
for all m > 0, n > 0.Hence evaluate (10,7) and
.
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| Life (in hours)
499.5 – 649.5 | Frequency
5 |
| 649.5 – 799.5 | 20 |
| 799.5 – 949.5 | 40 |
| 949.5 – 1099.5 | 81 |
| 1099.5 – 1249.5 | 57 |
| 1249.5 – 1399.5 | 34 |
| 1399.5 – 1549.5 | 13 |
| Total | 250 |
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