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Welcome to BUAD 310

Welcome to BUAD 310. Instructor: Kam Hamidieh Lecture 9, Wednesday February 12, 2013. Agenda & Announcement. Today : (Quick) Demo of StatCrunch on getting normal probabilities (will put our YouTube clip soon.) Finish up from last time, Chapters 10 & 12 Note:

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Welcome to BUAD 310

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  1. Welcome to BUAD 310 Instructor: Kam Hamidieh Lecture 9, Wednesday February 12, 2013

  2. Agenda & Announcement • Today: • (Quick) Demo of StatCrunch on getting normal probabilities (will put our YouTube clip soon.) • Finish up from last time, Chapters 10 & 12 • Note: • Homework 2 is due today by 5 PM. • Homework 3, due 12/19, should be up soon. • Be counted! Take the poll:https://docs.google.com/forms/d/17BRu7uDGCNfzMKAp6jgcxHXSQr-dfV2hnphcojnV-hA/viewform • We do not have class on Monday February 17th. BUAD 310 - Kam Hamidieh

  3. Exam 1 • It will be in class on Wednesday February 26, 2014. • It will cover everything up to the end of Wednesday February 19 lecture. • You’ll have 33 multiple choice questions & use Scantron. • You will get the coversheet and the scantron sheet a couple of days before the exam. • Practice problems will posted after Wednesday 2/19. • We’ll spend 2/24 doing review. • From our syllabus “You may bring a single handwritten sheet (both sides) containing formulas to each test …. No make-ups of tests will be given. You will receive a grade of zero for each missed test unless you have a written excuse from your doctor or the University.” BUAD 310 - Kam Hamidieh

  4. Last Time • X ~ N(μ, ), E(X) = μ, SD(X) =  and the empirical rule: μ-3 μ-2 μ- μ+ μ+2 μ+3 μ • X ~ N(μ, ) => (X – μ)/  ~ N(0,1) (standard normal) • z-score of x: z = (x – μ)/ • Note: z-scores can be applied to any data or RVs and not just normal but when data (or RV) is normal then we could make probability statements. • Calculating probabilities: empirical rule or table or software. Make use of symmetry! BUAD 310 - Kam Hamidieh

  5. A Word About Notation • X ~ N(μ, ) means X is a normal random variable with mean μ and standard deviation . • Your book also uses N(μ, 2). This means normal with mean μand variance 2. BUAD 310 - Kam Hamidieh

  6. Joint Behavior In this chapter we will study the joint behavior of two random variables: Given two random variables X and Y, do their probabilities depend on each other? What can we get out of knowing joint probabilities? Can we somehow summarize their joint behavior? Are the above questions useful in business? (Not fully answered but coming!) BUAD 310 - Kam Hamidieh

  7. Motivation Consider: X = Mean Temperature, Y = Electricity consumption BUAD 310 - Kam Hamidieh

  8. Motivation Consider: X = Microsoft returns, Y = IBM returns • Questions: • Are X and X related? • If yes, can we capture that information somehow? • Why care? BUAD 310 - Kam Hamidieh

  9. Motivating Example • Consider: X = Change in IBM stock, Y = Change in Microsoft stock • Changes are for the next day. • Note: This is an idealized example! BUAD 310 - Kam Hamidieh

  10. Joint Probabilities The joint probability distribution of two or more random variables is the probability of those random variables taking on specified values at the same time: P(X = x and Y = y) = p(x,y) Here for example:P(X = -5 and Y = 0) = 0.03P(X = 0 and Y = 0) = 0.62P(X = 5 and Y = -4) = 0.02 Marginal Distribution BUAD 310 - Kam Hamidieh

  11. Probability Rules & Joint Distributions • If you let { X = x } = A, and { Y = y } = B then all the probability rules we learned apply! • Example: { IBM is unchanged } = { X = 0 } = A{ Microsoft goes up } = { Y = 4 } = B P( IBM is unchanged or Microsoft goes up ) = P( X = 0 or Y = 4 ) = P( A or B ) = P(A) + P(B) – P( A and B) = 0.80 + 0.18 – 0.11 = 0.87 BUAD 310 - Kam Hamidieh

  12. Conditional Probabilities • Recall definition of condition probability:P(A|B) = P(A and B)/P(B), (P(B)>0) • Let P(X = x | Y = y) = P(X = x and Y = y) / P(Y = y) A A B B B BUAD 310 - Kam Hamidieh

  13. Independence of RVs • Recall independent events: P(A|B) = P(A) (P(A)>0) or P(A and B) = P(A) × P(B) • Now you can ask if two random variables are independent. • On random variables:X, Y are independent  P(X = x and Y = y) = P(X = x) × P(Y = y) for all x,yorX, Y are independent  P(X = x | Y = y) = P(X = x) for all x,y(P(Y=y) > 0) BUAD 310 - Kam Hamidieh

  14. Example • Are IBM and Microsoft changes independent? • If yes, it means that the probabilities for the Microsoft changes does not get affected by the IBM changes. P(X = 0 and Y = 0) ?=? P(X = 0) × P(Y = 0) 0.62 ≠0.80 × 0.67 NOT Independent! BUAD 310 - Kam Hamidieh

  15. In Class Exercise 1 We will use the IBM (X) and Microsoft (Y) Example • Express the event Microsoft stock goes down by 4 and find the probability of this event. • What is the probability that Microsoft is unchanged given that IBM goes up by 5? • What is the probability that the Microsoft is unchanged or IBM is unchanged? BUAD 310 - Kam Hamidieh

  16. We Need More… • Suppose we want to form a new random variable as follows: W = X + Y. • What is W? Think of buying 1 share of IBM and 1 share of MSFT. W is the random variable that represents the possible changes in your portfolio. • Suppose we want to find: E[W] and SD(W). Why would we care about these quantities?We want to know about the expected value of our portfolio and its uncertainty! BUAD 310 - Kam Hamidieh

  17. Finding Distribution of a New RV SD(W) = SD(X + Y) = 3.82 E(W) = E(X + Y) = 0.22 BUAD 310 - Kam Hamidieh

  18. New Rules • Addition rule for expected value of sums of random variable: E(X + Y) = E(X) + E(Y) • Addition rule for variance of independentrandom variables:Var(X + Y) = Var(X) + Var(Y) • Addition rule for variances:Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X,Y) ? BUAD 310 - Kam Hamidieh

  19. Covariance & Correlation • The covariance between random variables is the expected value of the product of deviations from the mean: • The correlation between two random variables is just covariance divided by the product of their standard deviations: • Note the covariance is part of correlation’s formula. BUAD 310 - Kam Hamidieh

  20. More on Correlation • Correlation will be our preferred number over covariance. Covariance can be made arbitrarily large with change of units. • Correlation attempts to compress the relationship between two random variables into a single number. • You can show that: -1  Corr(X,Y)  +1 • It is a good measure of linear dependence of two random variables. BUAD 310 - Kam Hamidieh

  21. Even More on Correlation • Values near: • 1 mean strong positive linear relationship • 0 mean week linear relationship • -1 mean strong negative linear relationship • Correlation is unit-less. This is a big advantage. BUAD 310 - Kam Hamidieh

  22. Some Terms • If Cov(X,Y) ≠ 0 or alternatively Corr(X,Y) ≠ 0 , then we say X and Y are correlated. • If Cov(X,Y) = 0 or alternatively Corr(X,Y) = 0 , then we say X and Y are uncorrelated. BUAD 310 - Kam Hamidieh

  23. Intuition Read the math equation: Take a “weight average” using Joint density Z-scores of Y values Z-scores of X values × BUAD 310 - Kam Hamidieh

  24. Covariance Calculation Cov(X,Y) = 2.19, Corr(X,Y) = 2.19/(sqrt of (4.99×5.27))=0.43 Recall: SD(W) = SD(X + Y) = 3.82 Var(W) = Var(X+Y) = Var(X) + Var(Y) + 2 Cov(X,Y) = (4.99) + (5.27) + 2 (2.19)=14.64 Now: SD(W) = square root of 14.64 = 3.82 BUAD 310 - Kam Hamidieh

  25. Dependence vs. Correlation • If X and Y are independent then they are uncorrelated so Cov(X,Y) = 0 and Corr(X,Y) = 0 • However if X and Y are uncorrelated then it does not imply the X and Y are independent! BUAD 310 - Kam Hamidieh

  26. In Class Exercise 2 • T or F: If X and Y are independent then SD(X+Y) = SD(X) + SD(Y). • T or F: If X and Y are independent then X and Y are uncorrelated. • T or F: If X and Y are uncorrelated then X and Y are independent. • T or F: If Cov(X,Y) = 0, then Corr(X,Y) = 0. • T or F: If Cov(X,Y) = 2, SD(X) = 4, SD(Y) = 2, then Corr(X,Y) = 0.25. BUAD 310 - Kam Hamidieh

  27. Correlation & Dependence There is a relationship between X and Y but the correlation is near zero! BUAD 310 - Kam Hamidieh

  28. Some Comments Consider two random variables X and Y: • E[X] tells me what I expect X’s values to be. If you asked me to give you a single number to summarize the information about X, it is E[X]. • SD(X) tells me about the variability (spread or uncertainty) about X. If is it large then X values can be “all over the place” relative to its mean. If it is small then on average X values stay close to E[X]. • Note: if I have the actual distribution of X, then I have ALL the information about X because I can calculate E[X], SD(X), Median of X, etc. • Same things can be said about Y. BUAD 310 - Kam Hamidieh

  29. Some Comments • X + Y: a new random variable • E[X + Y] tell me what I expect to X + Y to be. • SD(X+Y) tells me about the variability (spread or uncertainty) about X + Y. Similar to SD(X). • Cov(X,Y) tells me about the relationship between X,Y. Usually used as an intermediate number for other calculations. It’s an input to correlation Corr(X,Y). • Corr(X,Y) tells me about the relationship between X, Y in a “much nicer way”. This is the most commonly quoted measure of association between two random variables. It must be used with caution! • Note: if I have the joint distribution of X and Y, then I have ALL the information about X and Y. We can calculate say E[X + Y]. BUAD 310 - Kam Hamidieh

  30. In Class THINKING 3!!! Discuss: If the distribution of a random variable X and the joint distributions of X and Y are so useful then why do we care about things like the expected values and standard deviations and correlations, etc? BUAD 310 - Kam Hamidieh

  31. Something to Look Forward to…. • Going back to the IBM & MSFT example, we assumed that we knew the marginal and joint distributions. • We’ll have to slightly change the formulas for the correlation when we do not know the actual distributions. We’ll have to estimate the correlation. BUAD 310 - Kam Hamidieh

  32. Next Time • Continue with some bits of Chapters 12 and 10, and start statistical inference. BUAD 310 - Kam Hamidieh

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