Random Variables Part 2 1 A P S T A T I S T I C S C H A P T E R 1 5 The theory of probabilities is simply the science of logic quantitatively treated. Charles Saunders Peirce (1839-1914)
Behavior of Random Variables 2 Adding or subtracting a constant from data shifts the mean, but does not change the variance. The same is true for random variables: mean of X E X c E X c Var X c Var X variance of X
Behavior of Random Variables 3 Multiplying each value in a set of data by a constant multiplies the mean by that constant, and multiplies the variance by the square of the constant. The same goes for random variables: E a X a E X Var a X a Var X 2
Behavior of Random Variables 4 Adding or subtracting random variables adds or subtracts their expected values (means): E X Y E X E Y
Behavior of Random Variables 5 Adding or subtracting independent random variables causes their variances to add. Var X Y Var X Var Y This is sometimes referred to as the Pythagorean Theorem of Statistics.
Behavior of Random Variables 6 Example. Given 2 independent random variables X and Y, find the mean and the standard deviation for each of the following: 1. X 15 2. 1.5Y 3. 4. 3X X X X 1 2 3 X Y Mean SD 50 12 20 5
Behavior of Random Variables 7 Example. Given 2 independent random variables X and Y, find the mean and the standard deviation for each of the following: 5. 6. Y Y Y 1 2 X 7. 2X 3Y X Y Mean SD 50 12 20 5
Example 8 Suppose the time it takes a customer to get and pay for seats at the ticket window of a baseball park is a random variable with a mean time of 100 seconds and standard deviation of 50 seconds. When you get there, you find two people in line ahead of you. Expected value = mean 1. How long should you expect before it is your turn?
Example 9 Suppose the time it takes a customer to get and pay for seats at the ticket window of a baseball park is a random variable with a mean time of 100 seconds and standard deviation of 50 seconds. When you get there, you find two people in line ahead of you. 2. What is the standard deviation of your wait time? What assumption do you have to make in order to answer this question?
Two Types of Random Variables 10 A random variable has a numerical value based on the outcome of a random event. We use capital letters, like X, to denote a random variable. If only certain values of X may occur the random variable is discrete. If any value within an interval may occur, the random variable is continuous.
Continuous Random Variable 11 A continuous random variable may take any of an infinite number of values in an interval, so we can t show the probability model as a table of discrete values. Instead, we indicate characteristics such as the shape, center, and spread of the model.
Continuous Random Variable 12 Perhaps one of the most well-known models for continuous random variables is the Normal Distribution model: N,
Continuous Random Variable The sum or difference of two independent and Normally distributed variables will also be Normally distributed. 13 X ~ N, 1 1 Y ~ N, 2 2 X Y ~ N, 2 2 1 2 1 2
Continuous Random Variable 14 A company manufactures small stereo systems. At the end of the manufacturing process, the shipping process has two stages. The first stage, called packing, involves putting all of the individual components into the molded styrofoam form. The second stage, called boxing, is placing the packed pieces into a box for shipping. The time to complete the first stage can be modeled by a Normal model with mean of 9 minutes and standard deviation of 1.5 minutes. Stage 2 can also be modeled with a Normal model with mean of 6 minutes and standard deviation of 1 minute.
Continuous Random Variable The time to complete the first stage can be modeled by a Normal model with mean of 9 minutes and standard deviation of 1.5 minutes. Stage 2 can also be modeled with a Normal model with mean of 6 minutes and standard deviation of 1 minute. 1. What is the probability that packing two consecutive systems takes over 20 minutes? 2. What percentage of the stereo systems take longer to box than to pack? 15
Read Chap 15 Assignment Exercises #27-31 odd, 35, 39, 43, 47, 51, 55 16 www.causeweb.org John Landers