Today Today: Chapter 5 Reading: –Chapter 5 (not 5.12) –Exam includes sections 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

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

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

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

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

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

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

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

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

Variance The variance of a continuous random variable is:

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

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

Properties

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)

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

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

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