Advanced Placement Statistics Section 7.2: Means & Variance of a Random Variable EQ: How do you calculate the mean, standard deviation, and variance of a random variable?

mean of a data set variance of a data set Part I: Means and Variance of Random Variables Recall: Means and Variance of Samples mean of a data set variance of a data set

standard deviation of a data set

But do you ever have to calculate these by hand?

Ex. You and I play a betting game: Statistics Recall: Samples Ex. You and I play a betting game: We flip a coin and if it lands heads, I give you a dollar, and if it lands tails, you give me a dollar. On average, how much am I expected to win or lose? RECALL: Expected Value

On average, in the long run, I can expect to win $0 in this game. RECALL: Expected Value expected winnings = lose (-$1) half the time win ($1) half the time On average, in the long run, I can expect to win $0 in this game.

mean of DRVx “expected value” or “weighted average”

variance of DRVx

standard deviation of DRVx NOT ON YOUR FORMULA SHEET! Why are we using μ and σ for DRV mean and standard deviation? DRV are FINITE values and represent every value that COULD possibly occur for the random variable X. Therefore they are parameters.

Ex. Given the following probability distribution: Find P(X = 3). P(X = 3) = .1 Why or why not? NO, it’s weighted data. E(X) = μx = 2(.15) + 3(.1) + 4(.2) + 5(.2) + 6 (.35) = 4.5 On average, in the long run, the expected value for this distribution is 4.5 (whatever the units are).

Ex. Given the following probability distribution: .1 σx = 1.43

Day 43 Agenda: NO DG today

What are the default lists for 1st Var Stat ? Ex. Given the following probability distribution: .1 How can you use lists in your graphing calculator to find expected value? X  L1 P(X)  L2 Use 1st Var Stat Compare the summary statistics to our values for μx and σx . What do you notice about Sx? Explain. What are the default lists for 1st Var Stat ?

Do p. 486 #23 (will refer back to ex on notes 7.1) number of girls in a randomly selected family of 3 children E(X) = μx = 0(.125) + 1(.375) + 2(.375) + 3(.125) = 1.5

Law of Large Numbers --- the result of performing the same experiment a large number of times. According to the law, the “long-run average” of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. What does the proportion of heads begin to approach as the number of trials gets closer to 1000?

How large is “large enough”? More variability in DRV More trials needed to reach µ Assignment: p. 486 #24, 26, 28

RECALL: Linear Transformations in Algebra Part II: Transformations of Random Variables RECALL: Linear Transformations in Algebra original values for x and y a value of x must be multiplied by 8 to obtain the original y value each value of x is multiplied by 5 and decreased by 2 to obtain the original y value

Ex. Given this sample for a random variable X X = {12, 14, 16} NOTE: NOT WEIGHTED DATA!! E(X) = μx = 12+14+16 = 14 3

Multiply random variable X by 4 and increase by 3 Multiply random variable X by 4 and increase by 3. Call this random variable Y. 3 4 51 59 67 E(X) = μy = 51+59+67 = 59 3

Rules for Means and Variances of Transformed Random Variables: Refer to the previous example.

Refer to the previous example. WHAT IS MISSING?  “a” don’t PLAY  Refer to the previous example.

Refer to the previous example. WHAT IS MISSING?  “a” don’t PLAY  Refer to the previous example. NOTE: The standard deviation of 2X and -2X are both equal to 2σx because this is the square root of 4σ2x .

Combining these random variables means X + Y Rules for Means and Variances of Combined Random Variables: Combining these random variables means X + Y

Example for combining random variables: If they work on separate jobs, how many hours, on average, will they bill for completing both jobs?

40 hr. + 35 hr. = 75 hr. What is the variance and standard deviation for completing both jobs? 5 hr. + 4 hr. = 9 hr.

What is the variance and standard deviation for completing both jobs? 5 hr. + 4 hr. = 9 hr. REMEMBER:

THQ#4 due Friday at the beginning of class. TRANSFORMING DOES NOT MEAN COMBINING Assignment: p. 486 #24, 26, 28 THQ#4 due Friday at the beginning of class.

We need to finish the second free response question for Media Center Ch 6.