Welcome to BUAD 310

1. Welcome to BUAD 310 Instructor: Kam Hamidieh Lecture 7, Wednesday February 5, 2014

2. Agenda & Announcement • Today: • Finish up Chapter 9 • We will do Chapter 10 & 12, a bit out Chapter 4 (Pages 63-65 which we had skipped.) • For now we skip Chapter 10. • HW 2 is due next Wednesday February 12th, 5 pm. BUAD 310 - Kam Hamidieh

3. FYI Interesting video:Your daughter is pregnant! BUAD 310 - Kam Hamidieh

4. From Last Time • Joint probabilities:Probabilities of the two or more variables taking on some values at the same time • Marginal probabilities:As far as we are concerned, this is just synonymous with probability model for a variable. The word “marginal” is used to emphasize its distinction from the joint. BUAD 310 - Kam Hamidieh

5. From Last Time • Probability trees:A method to enumerate and display all the possible combination of multiple events. Often used to find P(A|B) from P(B|A). • Random variable:Basically a variables (has *numbers* associated with it) but its values have probabilities. • Probability distribution of a random variable:A listing of the possible values of a random variable and their probabilities. BUAD 310 - Kam Hamidieh

6. Caution! • Don’t confuse P(A|B) with P(B|A). • Don’t confuse P(A ∩ B) with P(A|B). • Which is more useful in real life: P( A ∩ B ) or P(A|B)? • Recall if P(A|B) = P(A) then A and B are independent. BUAD 310 - Kam Hamidieh

7. Motivation for Density • Sometimes the overall pattern of the data is so regular that we can describe it by a smooth curve. • The curve is a mathematical model for the distribution. BUAD 310 - Kam Hamidieh

8. Density • Continuous random variable:the outcome can be any value in an interval or collection of intervals. • The probability distribution of continuous random variables are often expresses in terms of a density function. • Density function or curve for a continuous random variable X is a curve such that the area under the curve over an interval equals the probability that X is in that interval. P(a X  b) = area under density curve over the interval between the values a and b. • Two conditions on a proper density: • The density has to be always greater than or equal to zero. • The total area under the density has to be 1. BUAD 310 - Kam Hamidieh

9. Density • If X is a continuous random variable, P(a <X<b) is the area under the density curve above the interval between a and b • Note: all continuous probability distributions assign probability zero to every individual outcome: P(X= any single value) = 0 BUAD 310 - Kam Hamidieh

10. Uniform Density • On the right is a uniform density curve for the variable X. • The probability that X falls between 0.3 and 0.7 is the area under the density curve for that interval: P(0.3 ≤ X ≤ 0.7) = (0.7 – 0.3)×1 = 0.4 • In general if X is uniform on [a,b], then the density curve is BUAD 310 - Kam Hamidieh

11. Uniform Density • The probability of an interval is the same whether boundary values are included or excluded:P(0 ≤ X ≤ 0.5) = (0.5 – 0)×1 = 0.5P(0 < X < 0.5) = (0.5 – 0) ×1 = 0.5P(0 ≤ X < 0.5) = (0.5 – 0) ×1 = 0.5 • A more complicated probabilityP(X < 0.5 or X > 0.8) = P(X < 0.5) + P(X > 0.8) = 1 – P(0.5 < X < 0.8) = 0.7 BUAD 310 - Kam Hamidieh

12. In Class Exercise 1 A density curve in form of a triangle is shown below: • Verify by geometry that the area under this curve is 1. • Find P(X < 1). Sketch the curve, shade the area that represents the probability, then find that area. Do this for (3) and (4) too. • Find P(X < 0.5). • Find P(X > 0.5). BUAD 310 - Kam Hamidieh

13. Properties of Expected Values & SD • Let c be any constant: E(X ± c) = E(X) ± c SD(X ± c) = SD(X) • Don’t memorize them! Just think them through! • Note: The above are true for any random variable. BUAD 310 - Kam Hamidieh

14. Properties of Expected Values & SD Add c = 7 E(X+7)  7 + 3.3 = 10.7 SD(X +7)  1.37 E(X)  3.3 SD(X)  1.37 BUAD 310 - Kam Hamidieh

15. Properties of Expected Values & SD Multiplying by a Constant (c) E(cX) = c E(X) SD(cX) = |c| SD(X) BUAD 310 - Kam Hamidieh

16. Properties of Expected Values & SD Multiply by c = 2 E(2X)  2 × 3.3 = 6.6 SD(2X)  2 × 1.37 =2.74 E(X)  3.3 SD(X)  1.37 BUAD 310 - Kam Hamidieh

17. In General Rules for Expected Values: if a and b are constants and X is a random variable, then: E(a + bX) = a + bE(X) SD(a + bX) = |b| SD(X) Var(a + bX) = b2Var(X) BUAD 310 - Kam Hamidieh

18. Example Let X = Stock price of 1 share of Apple 1 year from now. (It was $502 on 2/3/2014.) You can think of X as a random variable. Suppose you have high confidence that E[X] = $580 and SD(X) = $50. See:http://finance.yahoo.com/q/ao?s=AAPL+Analyst+Opinion • What do the above numbers (580 and 50) mean? • What kind of meaning can you attach to 100X + $10,000? • What are E[100X + 10,000] and SD(100X + 10,000)? Their meanings? (Answers on the next slide…) BUAD 310 - Kam Hamidieh

19. Apple Example Continued • What does the above numbers mean?E[X] = $580. We expect that the Apple stock price will be $580 one year from now.SD(X) = $50. However, we expect the price to be within $50 of $580. (Note the higher SD, the higher the uncertainty!) • What kind of meaning can you attach to 100X + $10000?You can think of this as buying 100 shares of Apple and holding $10000 cash! • What are E[100X + 10000] and SD(100X + 10000)? Their meanings?E[100X + 1000] = 100(580) + 10000 = $68,000SD(100X + 10000) = 100(50) = $5000We expectthat our portfolio consisting of 100 shares of Apple and $10000 cash to be worth $68,000. We expect the portfolio value to be within $5000 of $68,000. BUAD 310 - Kam Hamidieh

20. Example • The average maximum temperature in Chicago for the month of July is 83 ̊F with a standard deviation of 1 ̊F. • Let X = average maximum Chicago temperature in July. • We know: E(X) = 83 ̊F, SD(X) = 1 ̊F. • What is this? (X – 32)/1.8 • Then BUAD 310 - Kam Hamidieh

21. In Class Exercise 2 (Problem 35) Let X be a random variable with μ = E(X) = 120 and  = SD(X) = 15. Find the mean and SD of each of these random variables that are defined from X: • X/3 • 2X – 100 • X + 2 • X - X BUAD 310 - Kam Hamidieh

22. Next Time • Chapters 10 & 12. BUAD 310 - Kam Hamidieh