Continuous Random Variables Lecture 36 Section 7.5.3 Mon, Mar 29, 2004

Continuous Probability Distribution Functions Continuous Probability Distribution Function (pdf) – A function such that the area under the curve over an interval a ≤ x ≤ b equal the probability that a ≤ X ≤ b. In other words, area = probability.

Example The TI-83 will return a random number between 0 and 1 if we enter rand and press ENTER. Let X be a random number from 0 to 1. Then X has a uniform distribution from 0 to 1.

Example The graph of the pdf of X. f(x) 1 x 1

Example What is the probability that the random number is at least 0.3?

Example What is the probability that the random number is at least 0.3? f(x) 1 x 0.3 1

Example What is the probability that the random number is at least 0.3? f(x) 1 Area = 0.7 x 0.3 1

Example What is the probability that the random number is between 0.3 and 0.8?

Example What is the probability that the random number is between 0.3 and 0.8? f(x) 1 x 0.3 0.8 1

Example What is the probability that the random number is between 0.3 and 0.8? f(x) 1 Area = 0.5 x 0.3 0.8 1

Example Now suppose we use the TI-83 to get two random numbers from 0 to 1, and then add them together. Let Y = the sum of the two random numbers. What is the pdf of Y?

Example The graph of the pdf of Y. f(y) 1 y 1 2

Example The graph of the pdf of Y. f(y) 1 Area = 1 y 1 2

Example What is the probability that Y is between 0.5 and 1.5? f(y) 1 0.5 1 1.5 2

Example The probability equals the area under the graph from 0.5 to 1.5. f(y) 1 y 0.5 1 1.5 2

Example Cut it into two simple shapes. The total area is 0.75. f(y) 1 0.5 1 1.5 2

Example Suppose we get 12 random numbers between 0 and 1 from the TI-83 and add them all up. Let X = sum of 12 random numbers from 0 to 1. What is the pdf of X?

Example It turns out that the pdf of X is approximately normal with a mean of 6 and a standard of 1. 6 7 8 9 5 4 3 N(6, 1) x

Example What is the probability that the sum will be between 5 and 7? P(5 < X < 7) = P(-1 < Z < 1) = 0.8413 – 0.1587 = 0.6826.

Example What is the probability that the sum will be between 4 and 8? P(4 < X < 8) = P(-2 < Z < 2) = 0.9772 – 0.0228 = 0.9544.

Experiment Use the TI-83 to see if we really do get a total between 5 and 7 about 68% of the time. Enter the expression sum(rand(12)) and press ENTER. Do it several times and see how often the total is between 5 and 7.

Experiment We should see a value between 5 and 7 about 68% of the time. We should see a value between 4 and 8 about 95% of the time. We should see a value between 3 and 9 nearly always (99.7%).

Exercise Roll a die 10 times and let X = the sum of the 10 numbers. Fact: X is approximately normal with mean 35 and standard deviation 5.4.

Exercise What is the probability that the sum will be at least 45? Between 20 and 50? 35 N(35, 5.4) x

Assignment Page 442: Exercises 64 – 69. Page 451: Exercises 104 – 106.