1. Today Today: Chapter 5 Reading: –Chapter 5 (not 5.12) –Exam includes sections 5.1-5.7 from Chapter 5 –Suggested problems: 5.1, 5.2, 5.3, 5.15, 5.25, 5.33, 5.38, 5.47, 5.53, 5.62

2. Median and Percentiles The median, m, of a continuous distribution is the point such that F(m)=1/2 The 100p th percentile of a continuous distribution is the point, x p, such that F(x p )=p

3. Example (5-27) Suppose Y has cdf F(y)=y 3 for 0<y<1 What is the density of Y? Find the median of this distribution Find the 65 th percentile What is the probability that Y falls between the 25 th and 65 th percentiles

4. Some Important Percentiles First Quartile: Second Quartile: Third Quartile:

5. Example (5-27) Suppose Y has cdf F(y)=y 3 for 0<y<1 What are the quartiles for this distribution?

6. Expected Value The mean (expected value) of a continuous random variable is:

7. Example (5-27) Suppose Y has cdf F(y)=y 3 for 0<y<1 Find the expected value of Y Find the expected value of Y 2

8. Expected Value The mean (expected value) of a function, g(X), of a continuous random variable is:

9. Example (5-27) Suppose Y has cdf F(y)=y 3 for 0<y<1 Find the expected value of Y 2 Find the expected value of Y-Y 2

10. Variance The variance of a continuous random variable is:

11. Example (5-34) Suppose X has cdf F(x)=x 4 for 0<x<1 Find the variance of X Find the standard deviation of X

12. Bivariate Distributions If X and Y are jointly distributed random variables. The joint distribution function is: The joint density is:

13. Properties

14. Example Suppose X and Y have joint pdf f(x,y)=4xy for 0<x<1, 0<y<1 Find Prob(0<x<.5,.5<y<.75)

15. Marginal distribution If X and Y are jointly distributed continuous random variables, their marginal distributions are:

16. Example Suppose X and Y have joint pdf f(x,y)=4xy for 0<x<1, 0<y<1 Find the marginal distribution of X Find the marginal mean of X and Y

17. Example Suppose X and Y have joint pdf f(x,y)=4xy for 0<x<1, 0<y<1 Find E(XY)