Multivariable Calculus, Ninth Edition by Ron Larson, Bruce H. Edwards

By Ron Larson, Bruce H. Edwards

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Example text

Show that if X ∼ ????(????, ???? 2 ) then for ???? ∈ { − 1, 1}, ????[max{????(e − K), 0}] = ????e X ????+ 21 ???? 2 ( Φ ????(???? + ???? 2 − log K) ???? ) ( − ????KΦ ????(???? − log K) ???? ) where K > 0 and Φ(⋅) denotes the cumulative standard normal distribution function. Solution: We first let ???? = 1, ∞ ????[max{eX − K, 0}] = ∫log K (ex − K) fX (x) dx ∞ = −1 1 e 2 (ex − K) √ ∫log K ???? 2???? ∞ = By setting ???? = ∫log K −1 1 e 2 √ ???? 2???? ( ) x−???? 2 +x ???? ( ) x−???? 2 ???? dx ∞ dx − K ∫log K −1 1 e 2 √ ???? 2???? ( ) x−???? 2 ???? x−???? and z = ???? − ???? we have ???? ∞ ????[max{eX − K, 0}] = ∫ log K−???? ???? ∞ 1 2 1 2 1 1 e− 2 ???? +????????+???? d???? − K e− 2 ???? d???? √ √ log K−???? ∫ 2???? 2???? ???? dx.

Minimum and Maximum of Two Correlated Normal Distributions. Let X and Y be jointly normally distributed with means ????x , ????y , variances ????x2 , ????y2 and correlation coefficient ????xy ∈ (−1, 1) such that the joint density function is fXY (x, y) = 2????????x ????y 1 √ − e 1 2(1−????2xy ) [ ( 1 − ????2xy x−????x ????x )2 ( −2????xy x−????x ????x ] )( y−???? ) ( y−???? )2 y y + ???? ???? y y . Show that the distribution of U = min{X, Y} is ????xy ????y ⎞ ⎛ ⎛ −u + ???? + ????xy ????x (u − ???? ) ⎞ x y ⎟ ????y ⎜ −u + ????y + ????x (u − ????x ) ⎟ ⎜ fU (u) = Φ ⎜ √ √ ⎟ fX (u) + Φ ⎜ ⎟ fY (u).

Similarly we can also show that ⎞ ⎛ ⎜ ???? − ????xy y ⎟ ????(????, ????, ????xy ) = f (y)Φ ⎜ √ ⎟ dy ∫−∞ Y ⎜ 1 − ????2xy ⎟ ⎠ ⎝ ???? 1 2 1 where fY (y) = √ e− 2 y and ????(????, ????, ????xy ) + ????(−????, ????, −????xy ) = Φ(????). 2???? ◽ 17. Bivariate Normal Distribution Property. Let X and Y be jointly normally distributed with means ????x , ????y , variances ????x2 , ????y2 and correlation coefficient ????xy ∈ (−1, 1) such that the joint density function is [ fXY (x, y) = 2????????x ????y 1 √ − e 1 2(1−????2xy ) ( x−????x ????x )2 ( −2????xy x−????x ????x ] )( y−???? ) ( y−???? )2 y y + ???? ???? y 1 − ????2xy y .

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