1. Economics 105: Statistics Any questions? Go over GH 3

2. Properties of Expected Values E(a + bX) = a + bE(X), where a and b are constants If Y = a + bX, then var(Y) = var(a + bX) = b 2 var(X)

3. Example Let C = total cost of building a pool Let X = days to finish the project C = 25,000 + 900X X P(X = x i ) 10.1Find the mean, std dev, and 11.3 variance of the total cost. 12.3 13.2 14.1

4. “Let’s Make a Deal” Problem Intro http://math.ucsd.edu/~crypto/Monty/montybg.html http://math.ucsd.edu/~crypto/Monty/montybg.html Applet http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html Let’s play.

5. Permutations and Combinations Need to count number of outcomes Number of orderings – x objects must placed in a row – can only use each once – x! = (x)(x-1)(x-2) … (2)(1) called “x factorial” Permutations – suppose these x ordered boxes can be filled with n objects – n > x – What is the number of possible orderings now? – Permutations of n objects chosen x at a time = n P x – n P x = n(n-1)(n-2) … (n-x+1) = n!/(n-x)!

6. Permutations and Combinations How many ways to arrange, in order, 2 letters selected from A through E? What if order doesn’t matter? Combinations n C x = n P x /x! = n!/ [(n-x)! * x!] Eight people (5 men, 3 women) apply for a job. Four employees are needed. If all combinations are equally likely to be hired, what is the probability no women will be hired?

7. The Binomial Distribution Binomial Poisson Probability Distributions Discrete Probability Distributions Hypergeometric Bernoulli

8. Binomial Distribution Binomial distribution is composed of repeated Bernoulli trials Let X 1, X 2, …, X N be Bernoulli r.v.’s, then B is distributed binomially Probability of x successes in N trials is where p is the prob of “success” on a given trial

9. Binomial Distribution Let B ~ binomial, with p = prob of success, N = number of trials Find E[B] and Var[B] … but first a couple more rules on the mathematics of expectations with more than 1 r.v.

10. 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:

11. Binomial Distribution Let B ~ binomial and now find E[B] and Var[B] McCoy’s Tree Service in Mocksville, NC removes dead trees from commercial and residential properties. They have found that 40% of their invoices are paid within 10 working days. A random sample of 7 invoices is checked. What is the probability that fewer than 2 will be paid within 10 working days?