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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) = b2var(X)
Example• Let C = total cost of building a pool • Let X = days to finish the project• C = 25,000 + 900X• X P(X = xi)
10 .1 Find the mean, std dev, and
11 .3 variance of the total cost.
12 .3
13 .2
14 .1
“Let’s Make a Deal” Problem
• Intro http://math.ucsd.edu/~crypto/Monty/montybg.html
•Applet http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html
• Let’s play.
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 = nPx
– nPx = n(n-1)(n-2) … (n-x+1) = n!/(n-x)!
Permutations and Combinations• How many ways to arrange, in order, 2 letters selected from A through E?• What if order doesn’t matter?
• CombinationsnCx = nPx/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?
The Binomial Distribution
Binomial
Poisson
Probability Distributions
Discrete Probability
Distributions
Hypergeometric
Bernoulli
Binomial Distribution• Binomial distribution is composed of repeated Bernoulli trials •Let X1, X2, …, XN 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
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.
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:
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?