Probability Distribution Functions Objectives By the end of this lesson you should be able to do the following 1 State the properties of a cumulative distribution function cdf 2 Build a table for a discrete model demonstrating the cdf 3 Calculate probabilities of events by moving correctly between pmf and cdf Background In the last lesson we presented a formula for a binomial probability model For each of m trials where we want exactly n successes b n m p n Cm p m 1 p n m Recall that p is probability of success hence the probability of failure is 1 p Your text uses a different notation for the n the binomial coefficient m combination multiplier n Cm Probability Distribution Functions Let s shift the appearance of things Instead of using m let s assign the number of successes the variable x 1 n x Then this calculation becomes the function of x f x p x 1 p n x This is just a probability mass function pmf We might see it also written this way just emphasize this n x n x f X x p 1 p x Example The Dallas Business journal reported on August 5 2008 that Dallas based Southwest Airlines ranked third for lowest rates of canceled flights with just 0 3 percent of its flights canceled in June 2008 What is the probability that 5 of 13 SWA flights scheduled to depart Sky Harbor at 8 00 am will be cancelled It s not to hard to find this answer its just hard to believe 13 0 003 5 1 0 003 13 5 3 10 10 Why is mine always the one they cancel 5 For x 5 f 5 Here s a critical fact you probably remember We will use it in just a minute n If f x is a probability mass function then 0 f x 1 and f x 1 i 1 1 Nothing has changed really It was always a function of m We just seems to recognize things with x as a function It s a Pavlovian response I guess Arizona State University Department of Mathematics and Statistics 1 of 3 Probability Distribution Functions Let s extend our last example a step further Example What is the probability that at least 11 SWA flights of 13 on the schedule are not cancelled The thinking is to identify the acceptable situations They are that 11 12 or 13 flights are not cancelled Since they cannot overlap we will add the probabilities Writing the notation is a chore but we should do it for practice Also notice that we are now focused on success not cancelled not failure cancelled The probability of success is p 0 997 13 11 2 13 12 1 0 997 0 003 0 997 0 003 11 12 f x 11 12 13 13 0 997 13 0 003 0 13 Suppose we just decided to pile up accumulate the probabilities starting at 0 on time through some k 13 on time We could write the sum of all probabilities from 0 k 13 this way k x 13 x x 0 997 1 0 997 13 x 0 This is the cumulative distribution function cdf for our specific problem If F x is a cumulative distribution function for the pmf f x then F X x f i i x This function F represents the accumulation of probabilities for all values of the random variable X at or below x Recall that 0 f x 1 So both f and F are positive values functions Further both f and F are always in the bounded interval 0 1 As a last detail if either f 0 or F 0 is zero you are talking about an impossible situation Notice that the accumulation has a capital letter This is intentional since it relates this to the distribution function f exactly in the way we used a capital F to talk about the Riemann sum for a function f or in the continuous situation the integral You might also notice that I could have written the result for my extended problem as 13 13 x 13 x x p 1 p x 11 Do not confuse this abbreviation of a calculation for the specific event x 11 12 13 with the cdf By f x 11 12 13 definition the cdf starts with the lowest value of the random variable 2 of 3 Arizona State University Department of Mathematics and Statistics Probability Distribution Functions Example 0 1 for x 1 2 Let s use a different discrete pmf Let f X x 0 2 for x 3 4 0 4 for x 5 Create the table for the cumulative distribution function F x This is simple List all the outcomes in a column List the probabilities in the next column Add a column for F x and progressively add the probabilities X f X x F x X 1 0 1 0 1 2 0 1 0 2 3 0 2 0 4 4 0 2 0 6 5 0 4 1 0 Total 1 0 Don t Now answer the following questions Pay particular attention to whether F or f is used 1 Find F X 3 This is strictly a CDF question Interpret it as What is the probability that a value of 3 or smaller is obtained It also says What is the cumulative probability at X 3 Looking at the F column on the X 3 row read the value as 0 4 2 Find f X 3 This is interpreted as Find the probability for the event where X is three or more So many ways to do it I would use f X 3 1 F X 2 1 0 2 0 8 only because it looks so elegant 3 Find f X 3 This is interpreted as Find the probability for the event where X is more than three I would use f X 3 1 F X 3 1 0 4 0 6 4 Find f 2 X 4 This is interpreted as Find the probability for the event where X is at least 2 but not more than 4 I would use f 2 X 4 F X 4 F X 1 0 6 0 1 0 5 You are equally correct to add f 2 X 4 f X 2 f X However when we get to LARGE tables that may be prohibitively difficult 3 f X 4 Don t you love this stuff Remember my first probability lesson We haven t learned anything not covered there with all this formalism The reason I used the formalistic approaches will be more obvious when we get to the continuous pmf s and continuous cdf s I hope you already see the similarity between integration formulas and these processes I just used Arizona State University Department of Mathematics and Statistics 3 of 3