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Are you Ready for the Quiz Today

Are you Ready for the Quiz Today. Yes No With the little help from a friend. Upcoming In Class. Quiz 3 Thursday – February 21 (HW4 and HW5) Sunday Homework 6 Exam 1 – March 7th. Chapter 16. Random Variables. Everything. More About Means and Variances.

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Are you Ready for the Quiz Today

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  1. Are you Ready for the Quiz Today • Yes • No • With the little help from a friend

  2. Upcoming In Class • Quiz 3 Thursday – February 21 • (HW4 and HW5) • Sunday Homework 6 • Exam 1 – March 7th

  3. Chapter 16 Random Variables

  4. Everything

  5. More About Means and Variances • Adding or subtracting a constant from data E(X ± c) = E(X) ± c Var(X ± c) = Var(X) • Multiplying each value of a random variable by a constant multiplies the mean by that constant and the variance by the square of the constant: E(aX) = aE(X) Var(aX) = a2Var(X)

  6. More About Means and Variances (cont.) • In general, • The mean of the sum of two random variables is the sum of the means. • The mean of the difference of two random variables is the difference of the means. E(X ± Y) = E(X) ± E(Y) • If the random variables are independent, the variance of their sum or difference is always the sum of the variances. Var(X ± Y) = Var(X) + Var(Y)

  7. For the following problems , • Let, • Mean of X = 60 • SD of X = 10 • Mean of Y = 10 • SD of Y = 2

  8. Find the mean and SD for the random variable 4X+2Y • 4*60+2*10 and 4*10 • 4*60+2*10 and 2*2 • 4*60+2*10 and 4*10+2*2 • 4*60+2*10 and sqrt(16*100+4*4)

  9. Find the mean and SD for the random variable 2X-5Y • 2*60-5*10 and sqrt(2*10-5*2) • 2*60-5*10 and sqrt(4*10-25*2) • 2*60-5*10 and sqrt(4*100-25*4) • 2*60-5*10 and sqrt(4*100+25*4)

  10. Find the mean and SD for the Random Variable X1+X2 • 60+60 and sqrt(100+100) • 60+60 and sqrt(10+10) • 60+60 and 100+100 • 60+60 and 10+10

  11. Egg Example • A grocery supplier believes that in a dozen eggs, the mean number of broken eggs is 0.5 with a SD of 0.2 eggs. • You buy 3 dozen eggs.

  12. How many broken eggs do you expect in the three cartons? • 0.5+0.5+0.5 • 0.5*0.5*0.5 • 0.5+0.2 • I need a probability model to answer this question

  13. What’s the SD of the number of broken egg when buying three cartons? • 0.2*0.2*0.2 • 0.2+0.2+0.2 • Sqrt(0.2+0.2+0.2) • Sqrt(0.22+0.22+0.22)

  14. What assumption did you make about the eggs? • The cartons are dependent • The number of broken eggs is a continuous random variable. • The number of broken eggs is a discrete random variable. • The cartons are independent of each other.

  15. *Correlation and Covariance • If X is a random variable with expected value E(X)=µ and Y is a random variable with expected value E(Y)=ν, then the covariance of X and Y is defined as • The covariance measures howXandYvary together.

  16. *Correlation and Covariance (cont.) • Covariance, unlike correlation, doesn’t have to be between -1 and 1. If X and Y have large values, the covariance will be large as well. • To fix the “problem” we can divide the covariance by each of the standard deviations to get the correlation:

  17. Insurance Policies • An insurance company estimates that it should make an annual profit of $130 on each homeowner’s policy, with a standard deviation of $4,000.

  18. What is the mean and SD if the company write 3 policies? • Mean=3*130SD=sqrt(3*4,000) • Mean=130+130+130SD=sqrt(3*4,0002) • Mean=3*130SD=3*4,000 • Mean=130+130+130SD=3*4,0002

  19. What is the mean and SD if the company write 10,000 policies? • Mean=10,000*130SD=sqrt(10,000*4,0002) • Mean=130+130+130SD=sqrt(10,000*4,0002) • Mean=10,000*130SD=10,000*4,000 • Mean=130+130+130SD=10,000*4,000

  20. Will the company be profitable? • No. The variance is larger than the mean. • Yes, $0 is 3.25 standard deviations below the mean for 10,000 policies • Yes, the expected value is greater than zero. • No. Catastrophes are far too unpredictable to expect a profit.

  21. What assumption did you make? • The annual profit on a policy is a continuous random variable • Losses are dependent • Losses are independent of each other • The annual profit on a policy is a discrete random variable.

  22. Cereal Bowls • Large Bowl • E(XL)=2.7 oz • SD(XL)=0.2 oz • Small Bowl • E(XS)=1.7 oz • SD(XS)=0.2 oz

  23. How much more cereal do you expect to eat by using the large bowl? What is the SD? • Mean=2.7-1.7SD=sqrt(0.22-0.22) • Mean=2.7-1.7SD=sqrt(0.22+0.22) • Mean=2.7+1.7SD=sqrt(0.22-0.22) • Mean=2.7+1.7SD=sqrt(0.22+0.22)

  24. Assuming normality, what is the prob. that the small bowl contain more cereal? • -3.57 • -3.25 • 0.0002 • 0.9998

  25. What are the mean and SD of the total amount in the two bowls? • Mean=2.7-1.7SD=sqrt(0.22-0.22) • Mean=2.7-1.7SD=sqrt(0.22+0.22) • Mean=2.7+1.7SD=sqrt(0.22-0.22) • Mean=2.7+1.7SD=sqrt(0.22+0.22)

  26. Assuming normality, what’s the prob. you poured more than 4.7 oz of cereal total? • 1.0714 • -1.0714 • .8577 • .1423

  27. Upcoming In Class • Quiz 3 Thursday – February 21 • (HW4 and HW5) • Sunday Homework 6 • Exam 1 – March 7th

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