(a) Evaluating the necessary integrals, verify the identities

\[\mu_{2}=\alpha+\beta \mu_{1} \quad \text { and } \quad \sigma_{2}^{2}=\sigma^{2}+\beta^{2} \sigma_{1}^{2}\]

on page 374 .

(b) Substitute \(\mu_{2}=\alpha+\beta \mu_{1}\) and \(\sigma_{2}^{2}=\sigma^{2}+\beta^{2} \sigma_{1}^{2}\) into the formula for the bivariate density given on page 374 , and show that this gives the final form shown on page 375 .

Transcribed Image Text:

f(x, y) = 1 = 1 e (x-1) 20 e . 1 2 e [y - (a + Bx)] 202 [y - (a + Bx)] 202 (x-141) + 20