VIEWS: 238 PAGES: 14 POSTED ON: 10/29/2012
Economics 105: Statistics • Any questions? • Go over GH 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(xi) = P(X = xi) – graph – table • Bernoulli distribution – graph & table ? • Cumulative distribution function Discrete Random Variable Summary Measures • Expected Value (or mean) of a discrete distribution (Weighted Average) N m X º E(X) = å X i P( X i ) i =1 X P(X) – Example: Toss 2 coins, 0 0.25 X = # of heads, 1 0.50 compute expected value of X: 2 0.25 E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0 Discrete Random Variable Summary Measures (continued) • Variance of a discrete random variable N s X = E[(X - mX ) 2 ] = å[X i - E(X)]2 P(X i ) 2 i=1 • Standard Deviation of a discrete random variable sX = s 2 X 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 Discrete Random Variable Summary Measures (continued) – Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1) s= å[X - E(X)] P(X ) i 2 i s = (0 -1) 2 (0.25) + (1-1) 2 (0.50) + (2 -1) 2 (0.25) = 0.50 = 0.707 Possible number of heads = 0, 1, or 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) = 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 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? • 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? The Binomial Distribution Probability Distributions Discrete Probability Distributions Bernoulli Binomial Poisson Hypergeometric Binomial Distribution • Binomial distribution is composed of repeated Bernoulli trials •Let X1, X2, …, XN be Bernoulli r.v.’s, then B is N B = å Xi distributed binomially i =1 •Probability of x successes in N trials is N -x P( B = x)= N Cx p (1 - p) x , x = 0,1,2,..., N 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: E(aX + bY) = aE( X ) + bE(Y ) • Variance of the sum of two random variables: Var(aX + bY) º s aX+bY = a2s X + b2s Y + 2abs XY 2 2 2 • Standard deviation of the sum of two random variables: s aX +bY = s aX +bY 2 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?