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

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)

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

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

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

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

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

Suddenly a UFO appears

The spaceship bathes you in bright light.

Then the spaceship, you and your car disappears. ?

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?

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

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

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?

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

Alien Abduction PDF Differentiate CDF to get PDF 010.1

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

Back to example Cdf pdf

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)

Recall Fundamental theorem of Calculus

Alien Abudction CDF PDF

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).

Alien Abduction Probability abducted in 1.3 to 2.4 miles

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

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

Problem Type I Given that x has pdf Find c

continued pdf of X is

Problem Type 2 Find cdf of X for previous problem

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

Problem Type 3 Find P(X≤1/3|X≤1/2)