Chapter 4 - Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009
Review Probability distribution function Useful results Relationship between the pdf and the cdf
Review Outline Review Probability distribution function In Chapter 3 we have seen that a continuous random variable is one that can take any possible value in a given interval. s People weight People height Distance between two cities
Review Probability distribution function Probability distribution functon Now if X is continuous random variable the probability distribution or probability density function (pdf) of X is a function f (x) such that P(a X b) = b a f (x)dx
Legitimate pdf Outline Review Probability distribution function A function is a legitimate pdf if it satisfies the following two conditions f (x) 0 X f (x)dx = 1
Important property Outline Review Probability distribution function Note that in the continuous case P(X = c) = 0 for every possible value of c. (why?) This has a very useful consequence in the continuous case: P(a X b) = P(a < X b) = P(a X < b) = P(a < X < b)
4.5 page 158 Review Probability distribution function
A continuous random variable X is said to have a uniform distribution on the interval [A, B] if the pdf of X is the following: 1 f (x) = B A, A X B 0, otherwise
If in a Friday quiz we denote with X the time that the first student will finish and X follows a uniform distribution in the interval 5 to 15 minutes. Find the probability distribution Find the P(X < 4), P(X < 12) and P(X > 7).
Useful results Relationship between the pdf and the cdf The cumulative distribution function (cdf) for a continuous random variable X is the following: F (x) = P(X x) = x f (y)dy
Useful results Relationship between the pdf and the cdf If in a Friday quiz we denote with X the time that the first student will finish and X follows a uniform distribution in the interval 5 to 15 minutes. Find the cumulative distribution function of X.
Outline Useful results Relationship between the pdf and the cdf In the continuous case is very useful to use the cdf to find probabilities using the formulas: P(X > a) = 1 F (a) P(a X b) = F (b) F (a)
Useful results Relationship between the pdf and the cdf If in a Friday quiz we denote with X the time that the first student will finish and X follows a uniform distribution in the interval 5 to 15 minutes. Find P(X > 7) and P(6 < X < 11).
Useful results Relationship between the pdf and the cdf Obtaining f (x) from F (x) If X is a continuous random variable with pdf f (x) and cdf F (x), then at every x at which the derivative of F (x), denoted with F (x), exists we have that F (x) = f (x). Prove this for the quiz example in the previous slide.
If p is a number between 0 and 1. Then the (100p)th percentile of the distribution of a continuous random variable X is denoted by η(p) and it satisfies the following: p = F ((η(p))) = η(p) f (y)dy
4.9 page 164
Median Outline By definition the median is the middle observation. When we have a continuous random variable the median is the same as the 50th percentile. So half the area under the curve is below the median (on the left) and half above the median (on the right).
Section 4.1 page 165 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17