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

2. Chapter 5 Continuous Random Variables Not all outcomes can be listed (e.g., {w 1, w 2, …,}) as in the case of discrete random variable Some random variables are continuous and take on infinitely many values in an interval E.g., height of an individual

3. Continuous Random Variables Axioms of probability must still hold Events are usually expressed in intervals for a continuous random variable

4. Example (Continuous Uniform Distribution) Suppose X can take on any value between –1 and 1 Further suppose all intervals in [-1,1] of length a have the same probability of occurring, then X has a uniform distribution on (-1,1) Picture:

5. Distribution Function of a Continuous Random Variable The distribution function of a continuous random variable X is defined as, Also called the cumulative distribution function or cdf

6. Properties Probability of an interval:

7. Example Suppose X~U(-1,1), with cdf F(x)=1/2(x+1) for –1<x<1 Find P(X<0) Find P(-.5<X<.5) Find P(X=0)

8. Example Suppose X has cdf, Find P(X<1/2) Find P(.5<X<3)

9. Distribution Functions and Densities Suppose that F(x) is the distribution function of a continuous random variable If F(x) is differentiable, then its derivative is: f(x) is called the density function of X

10. Distribution Functions and Densities Therefore, That is, the probability of an interval is the area under the density curve

11. Example Suppose X~U(0,1), with cdf F(x)=x for –1<x<1 What is the density of X? Find P(X<.33)

12. Properties of the Density

13. Example (5-16) Suppose X is a random variable and it is claimed that X has density f(x)=30x 2 (1-x) 2 for 0<x<1 Is f(x) a density? If yes, find the c.d.f. of X.

14. Example (5-15) Suppose X is a random variable and X has density f(x)=c(1-|x|) for |x|<1 and c is a positive constant Find c? Draw a picture of f(x) Find P(X>1/2)

15. Example X has an exponential density: Find F(x)

16. Example X has an exponential density: Find the density of Y=X 1/2

17. Transformations If Y=g(x) is a one-to-one function with inverse, g -1 (x), the density of Y can be obtained from the density of X as,

18. Example X has an exponential density: Find the density of Y=X 1/2

19. Example (5-21) Suppose X~U(-1,1) Find the density of Y=|X| Find P(-.5<Y<.75)