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Economics 105: Statistics

Economics 105: Statistics. Any questions? Go over GH 3 & 4. Discrete Random Variables. Take on a limited number of distinct values Each outcome has an associated probability We can represent the probability distribution function in 3 ways function ƒ(x i ) = P(X = x i ) graph table

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Economics 105: Statistics

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  1. Economics 105: Statistics Any questions? Go over GH 3 & 4

  2. Discrete Random Variables Take on a limited number of distinct values Each outcome has an associated probability We can represent the probability distribution function in 3 ways function ƒ(xi) = P(X = xi) graph table Bernoulli distribution graph & table ? Cumulative distribution function

  3. Discrete Random Variable Summary Measures • Expected Value (or mean) of a discrete distribution (Weighted Average) • Example: Toss 2 coins, X= # of heads, compute expected value of X: E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0 X P(X) 0 0.25 1 0.50 2 0.25

  4. 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

  5. Discrete Random Variable Summary Measures (continued) • Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1) Possible number of heads = 0, 1, or 2

  6. 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)

  7. Example Let C = total cost of building a pool Let X = days to finish the project C = 25,000 + 900X XP(X = xi) 10 .1 Find the mean, std dev, and 11 .3 variance of the total cost. 12 .3 13 .2 14 .1

  8. 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)!

  9. Permutations and Combinations How many ways to arrange, in order, 2 letters selected from A through E? What if order doesn’t matter? Combinations nCx = 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?

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

  11. 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

  12. 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.

  13. 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:

  14. 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?

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