Page 239 - Probability and Statistical Inference

216 4. Functions of Random Variables and Sampling Distribution

Here, r is defined as the Pearson correlation coefficient
or simply the sample correlation coefficient.

By separately using the marginal distributions of X , ..., X and Y , ..., Y , we
n
1
n
1
can right away claim the following sampling distributions:

The joint distribution of is not very difficult to obtain by means of
transformations. We leave this out as the Exercise 4.6.8.

Alternately, note, however, that any linear function of and is also a
linear function of the original iid random vectors (X , Y ), i = 1, ..., n. Then, by
i
i
appealing to the Definition 4.6.1 of a multivariate normal distribution, we can
immediately claim the following result:

How does one find ρ*? Invoking the bilinear property of covariance from
Theorem 3.4.3, part (iii), one can express Cov as

so that the population correlation coefficient between and is simplified

to n ρσ σ / .
–1
1 2
The distribution of the Pearson correlation coefficient

is quite complicated, particularly when ρ ≠ 0. Without explicitly writing down
the pdf of r, it is still simple enough to see that the distribution of r can not
depend on the values of µ , and . To check this claim, let us
1
denote U = (X – µ )/σ , V = (Y – µ )/σ , and then observe that the random
i
2
2
i
i
1
i
1
vectors (U , V ), i = 1, ..., n, are distributed as iid N (0, 0, 1, 1, ρ). But, r can
2
i
i
be equivalently expressed in terms of the U ’s and V ’s as follows:
i i
From (4.6.8), it is clear that the distribution of r depends only on ρ.