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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009
Exercise 7: Joint distribution 11/10/2009
1. Covariance. For random variables X , Y and Z , you are given V ar(Z ) = 2, Cov (X, Y ) = −1, Cov (X, Z ) = 0, Cov (Y, Z ) = 1. X + 2Y and Y + 2Z . that V ar(X ) = V ar(Y ) = 1, Find the covariance between 2. Marginal, not rectangular. Let X and Y be continuous random variables with joint density function f (x, y ) =
Let 15y 0 Y for x2 ≤ y ≤ x, otherwise. g? g be the marginal density function of . What is 1 3. Uniform. Random variables X and Y are jointly distributed on the region 0 ≤ y ≤ x, y ≤ 1, x ≤ 1. • • • • • The joint distribution has a constant density over the entire region. Find: The joint densities of X and Y . The marginal densities of X and Y ≤ . The conditional densities of The probabilities Try a new region: X given Y = y, and Y given X = x. P (X > 1 2 ),P (Y 1 2 ), and P (X + Y ≤ 1). 0 ≤ y ≤ x, y ≤ 1, x ≤ 2 2 4. Joint→conditional. Let X and Y be continuous random variables with joint density function f (x, y ) =
Calculate
1 P (Y < X X = 3 ). 24xy 0 for 0 < x < 1, 0 < y < 1 − x, otherwise. 5. Conditional→joint. A company will experience a loss the interval (0, 2 X that is uniformly distributed be tween 0 and 1. The company pays a bonus to its employees that is uniformly distributed on − X ), which depends on the amount of the loss that occurred. Find the expected amount of the bonus paid. 3 ...
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 Fall '08
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