3.6 Practice Prelim Questions

3.6 Practice Prelim Questions - 20 A random variable X has...

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Review Problems for Prelim 3 Partial Derivatives Find ∂f/∂x and ∂f/∂y 1. x - y 2 2. (3 + 2 xy 2 ) 3 3. ln( x 2 + y 3 ) 4. y 2 e x 2 y 5. x 2 y x 2 + y 2 Find 2 f/∂x 2 and 2 f/∂y∂x 6. x 4 + x 2 y 2 + xy 3 + y 4 7. x y - y x 8. xe y/x Definite Integrals (includes Improper Integrals) Volume problems turn into these after you use V = b a πf ( x ) 2 dx 9. 25 4 1 x dx 10. 0 x (1 + x ) - 5 / 2 dx 11. 3 2 8 x ( x 2 - 4) 3 dx 12. 2 1 2 x 2 +1 x dx 13. 2 0 2 x x 2 +1 dx 14. 2 0 6 x 3 2 x 2 +1 dx 15. 4 1 ln x x dx 16. e 1 ln x x dx 17. 1 e - 2 x x dx 18. 0 x 3 e - x 2 / 2 dx Probability Density Problems 19. A random variable X has density function 12 x 2 (1 - x ), Y has density function 12 x (1 - x ) 2 . Compute their means and variances. If you do this correctly then the means will add to 1 and the variances will be the same, since X and 1 - Y have the same distribution.
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Unformatted text preview: 20. A random variable X has probability density f ( x ) = (1 / 7)(1 + 2 x-1 / 2 ) for 1 ≤ x ≤ 4. Find the mean and variance of X . 1 21. Suppose that baseball player salaries in units of hundreds of thousands of dollars have a probability density f ( x ) = cx-9 / 4 for x ≥ 1. (a) Find the constant c to make this a probability density. (b) What fraction of players make more than 10 million dollars (100 hundred thousand) (c) Find the average players salary. (d) Find the variance. 2...
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