1. Means and Variances of Random Variables

2. Activity 1 : means of random Variables To see how means of random variables work, consider a random variable that takes values {1,1,2,3,5,8}. Then do the following: 1.Calculate the mean of the population: 2.Make a list of all the sample of size 2 from this population. You should have 15 subsets of size 2 3.Find the mean of the 15 x-bar in the third column and compare the result with the population mean. 4.Repeat steps 1-3 for a different (but still small) populations of your choice. Now compare your result with each other. 5.Write a brief statement that describes what you discovered. 1.Calculate the mean of the population: 2.Make a list of all the sample of size 2 from this population. You should have 15 subsets of size 2 3.Find the mean of the 15 x-bar in the third column and compare the result with the population mean. 4.Repeat steps 1-3 for a different (but still small) populations of your choice. Now compare your result with each other. 5.Write a brief statement that describes what you discovered.

3. Mean of a random Variable The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take.

4. example A Tri-State Pick 3 game in New Hampshire, Maine and Vermont let you choose three-digit number and the state chooses three-digit winning number at random and pays you $500 if your number is chosen. Because there are 1000 three digit numbers, you have a probability of 1/1000 of winning. Taking X to be the amount your ticket pays you, the probability distribution of X is:

5. What is the average payoff from the tickets? Payoff X$0$500 Probability0.9990.001 ($0 + $500)/2 = $250 The long-run average pay off is: $500(1/1000) + $0(999/1000) = $0.50

6. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The common symbol for the mean (also known as the expected value of X) is, formally defined by The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The common symbol for the mean (also known as the expected value of X) is, formally defined by The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.

7. Benford’s Law Calculating the expected first digit First digit X123456789 Probability1/9 First digit V123456789 Probability 0.3010.1760.1250.0970.0790.0670.0580.0510.046 Find the of the X distribution y= 3.4441 Find the of the V distribution x= 5

8. y= 3.4441

9. Variance of discrete Random Variables The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by: The standard deviation is the square root of the variance.

10. Gabby Sells Cars: Gabby is a sales associate at a large auto dealership. She motivates herself by using probability estimates of her sales. For sunny Saturday in April, she estimates her car sales as follows: Gabby is a sales associate at a large auto dealership. She motivates herself by using probability estimates of her sales. For sunny Saturday in April, she estimates her car sales as follows: Cars sold:0123 Probability0.30.40.20.1 Let’s find the mean and variance of X

11. Cars sold: X0123 Probability: P0.30.40.20.1 00.30.0(0-1.1) 2 (0.3) =0.363 10.4 (1-1.1) 2 (0.4) =0.004 20.20.4(2-1.1) 2 (0.2) =0.162 30.10.3(3-1.1) 2 (0.1) =0.361 1.10.890

12. Law of Large number

13. Rule for Means

14. Rules for Variances

15. Example: Cars sold: X0123 Probability: P0.30.40.20.1 Trucks and SUV: Y012 Probability: P0.40.50.1 Mean of X: 1.1 cars Mean of Y: 0.7 T’s & SUV’s Mean of Y: 0.7 T’s & SUV’s At her commission rate of 25% of gross profit on each vehicle she sells, Linda expects to earn $350 for each cars sold and $400 for each truck and SUV’s sold. So her earnings are: At her commission rate of 25% of gross profit on each vehicle she sells, Linda expects to earn $350 for each cars sold and $400 for each truck and SUV’s sold. So her earnings are: Z= 350 X + 400Y Z= 350 X + 400Y Combining rule 1 and 2 her mean earnings will be: Uz= 350 U X + 400 U Y Uz= 350 (1.1) + 400 (.7) = $665 a day