Business 90: Business Statistics Professor David Mease Sec 03, T R 7:30-8:45AM BBC 204 Lecture 16 = More of Chapter “Some Important Discrete Probability Distributions” (SIDPD) Agenda: 1) Go over solutions to Midterm Exam 2) Reminder about Homework 6 (due Tuesday 4/13) 3) Lecture over more of Chapter SIDPD

Homework 6 – Due Tuesday 4/13 1) Read chapter entitled “Some Important Discrete Probability Distributions” but only sections 1-3. 2) In that chapter do textbook problems 3, 4, 14, 15 and 20 (but skip part g in 20) 3) Stock X has a mean of $50 and a standard deviation of $10. Stock Y has a mean of $100 and a standard deviation of $20. Find the mean and standard deviation of buying one share of each A) If they are independent (so the covariance is 0) B) If the covariance is 30 C) If the covariance is -30

Some Important Discrete Probability Distributions Statistics for Managers Using Microsoft® Excel 4th Edition Some Important Discrete Probability Distributions

Chapter Goals After completing this chapter, you should be able to: Compute and interpret the mean and standard deviation for a discrete probability distribution Explain covariance and its application in finance Use the binomial probability distribution to find probabilities Describe when to apply the binomial distribution

Introduction to Probability Distributions Random Variable Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable (This Chapter) (Next Chapter)

Discrete Probability Distributions A discrete probability distribution is given by a table listing all possible values for the random variable along with the corresponding probabilities. The appropriate chart to display it is a bar chart (which has gaps, unlike a histogram).

In class exercise #64: A box contains two $1 bills, one $5 bill and one $20 bill. You reach in without looking and pull out a single bill. Give the probability distribution and bar chart for the amount of money you pull out.

In class exercise #65: A box contains two $1 bills, one $5 bill and one $20 bill. Many people reach in without looking and each pull out a single bill and put it back. On average, how much money would you expect each person to get? How much money would you personally be willing to pay to play this game once?

Discrete Random Variable Summary Measures Expected Value (or mean) of a discrete distribution (Weighted Average)

In class exercise #66: A fair coin is to be tossed two times. A) Give the expected number of tails. B) Give the variance for the number of tails. C) Give the standard deviation for the number of tails.

Discrete Random Variable Summary Measures (continued) Variance of a discrete random variable Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X Xi = the ith outcome of X P(Xi) = Probability of the ith occurrence of X

In class exercise #67: Shares of stock X and stock Y each cost $100 dollars per share. Your advisor estimates there is a 20% probability that in one year a share of stock X will be worth $90 and a share of stock Y will be worth $130, a 40% probability X will be worth $100 and Y will be worth $100, and 40% probability X will be worth $130 and Y will be worth $85. Compare the following three investment options in terms of mean, variance and standard deviation. 1) One share of X 2) One share of Y 3) One share of each

The Covariance The covariance measures the strength and direction of the linear relationship between two variables I will not ask you to compute it, but here is the formula where: X = discrete variable X Xi = the ith outcome of X Y = discrete variable Y Yi = the ith outcome of Y P(XiYi) = probability

In class exercise #68: Would you guess that the covariance between stock X and stock Y to be positive or negative. Why?

The Sum of Two Random Variables Expected Value of the sum of two random variables: Variance of the sum of two random variables: Standard deviation of the sum of two random variables: E(X+Y) = E(X) + E(Y)

In class exercise #69: The covariance between stock X and stock Y is -240. Use this information and the formulas on the previous slide to check your answer for the expected value and variance when you buy one share of stock X and one share of stock Y.