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14.2 Exercise 7

Avg difficulty 2.5

(See Exercise 16 from " Problems On Random Vectors and Joint Distributions", Exercise 26 from "Problems on Mathematical Expectation", and Exercise 26 from "Problems on Variance, Covariance, Linear Regression"). fXY(t,u)=12t2uf_{X Y} \left(t , u\right) = 12 t^{2} u on the parallelogram with vertices

( - 1 , 0 ) , ( 0 , 0 ) , ( 1 , 1 ) , ( 0 , 1 ) \left(- 1 , 0\right) , \left(0 , 0\right) , \left(1 , 1\right) , \left(0 , 1\right)

The regression line of Y on X is u=(4t+5)/9u = \left(4 t + 5\right) / 9.

f X ( t ) = I [ - 1 , 0 ] ( t ) 6 t 2 ( t + 1 ) 2 + I ( 0 , 1 ] ( t ) 6 t 2 ( 1 - t 2 ) f_{X} \left(t\right) = I_{\left[- 1 , 0\right]} \left(t\right) 6 t^{2} \left(t + 1\right)^{2} + I_{\left(0 , 1\right]} \left(t\right) 6 t^{2} \left(1 - t^{2}\right)

Pfeiffer, Applied Probability, 14.2 Problems on Conditional Expectation, Regression, exercise 7

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Adapted from Applied Probability © 2009 Paul E. Pfeiffer. Licensed under CC BY 3.0. Source: Applied Probability. Imported at revision d7a0420.