1. Lecture 10: Continuous RV Probability Theory and Applications Fall 2005 September 29 Everything existing in the universe is the fruit of chance. Democritus

2. To Come More on MGF after we learn continuous distributions. Test 10/2 –Covers materials up through this lecture –Bring Calculator –Bring one page of notes (both sides fine) –Sample exam on web (sorry no answers)

3. WARNING To properly specify a CDF you give it for all possible values.

4. RIGHT F(x)x 0x<1 1/551≤x<2 5/552 ≤ x<3 14/553 ≤ x<4 30/554 ≤ x<5 55/555 ≤ x

5. WRONG WRONG WRONG WRONG F(x)x 00 1/551 5/552 14/553 30/554 ≤ x<5 55/555

6. Outline Motivating Example for CRV Continuous Random Variables Sample types of problems

7. Imagine…. Driving down on a 10 mile stretch of highway near Roswell New Mexico.

8. Suddenly a UFO appears

9. The spaceship bathes you in bright light.

10. Then the spaceship, you and your car disappears. ?

11. Alien Abduction Problem Imagine Fox and Mulder are driving down a 10 mile stretch of highway and they will be abducted by aliens stretch of highway. What is the probability they will be abducted in the first 10 miles assuming that their chance of getting abducted as any point of the road is equally likely?

12. Discrete Version There are mile markers that divide highway into 10 segments. Let X={1,2,..,10} be the probability you vanish after x-1 and up to mile marker x. X is discrete uniform. P(X=x)=1/10 x=1,..,10 Note and sketch CDF

13. Continuous Version Let X needs to be a real random variable since we could disappear at about 3.2 miles and that is different than 3.9 miles. Uniform assumption 010 0 1

14. Looks good Probability they disappear in the first half P(X≤5)=5/10=1/2 Seem like right cdf. What would the “pdf” be?

15. Probability of small interval Probably disappear between point a and b = F(b)-F(a). Probability disappear on a very small interval. Let by fundamental theorem of calculus

16. Alien Abduction PDF Differentiate CDF to get PDF 010.1

17. Continuous RV X is a continuous R.V., if and only if F(x)=P(X≤x) is a continuous function from the reals to [0,1] If F(x) is an integral of some function f(x)≥0 of the form then f(x) is called a probability density function p.d.f

18. Back to example Cdf pdf

19. PDF If F(X) is the cdf of a random variable and F(x) is an integral of some function of the form then is called the probability density function (pdf)

20. Recall Fundamental theorem of Calculus

21. Alien Abudction CDF PDF

22. Probability of Event Let X be a continuous R.V. with cdf F(x) and pdf f(x). Let A be an event (subset of R).

23. Alien Abduction Probability abducted in 1.3 to 2.4 miles

24. Alien Abduction Probability abducted at 1.3 miles The probability X=x for any x is 0!!

25. Note Let X be a R.V. with pdf f(x)

26. Problem Type I Given that x has pdf Find c

27. continued pdf of X is

28. Problem Type 2 Find cdf of X for previous problem

29. Problem Type 3 Find P(1/4<x<4) Using cdf F(4)-F(1/2)=1-(3/4-1/4)=1/2 Using pdf

30. Problem Type 3 Find P(X≤1/3|X≤1/2)