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Unformatted text preview: Stat 104: Quantitative Methods for Economists Class 12: Random Variables, Part II 1 What were we talking about? s Random variables s Mean, variance s Linear transformation rule 2 Rules for Expectation and Variance Let X be a random variable with mean μ and variance σ 2. Let a and b be any constant fixed numbers. Define the random variable W=a+bX. 3 Then Two other fascinating rules: E(a)=a and Var(a)=0. 2 2 ( ) ( ) ( ) ( ) ( ) W X X E W E a bX a bE X a b Var W Var a bX b μ μ σ = = + = + = + = + = Example s An author receives from a publisher a contract, according to which she is to be paid a fixed sum of $10,000 plus $1.50 for each copy of her book sold. s Her uncertainty about total sales of the book can be presented y ndom ariable ith ean 00 nd represented by a random variable with mean 30,000 and standard deviation 8,000. s Find the mean and standard deviation of the total payments she will receive 4 Solution s Payment =10000+1.5*X, where X = # of books sold s This is the a+bX framework with a=10000 nd b=1.5 and b=1.5 s E(Payment) = 10000+1.5E(X), E(X)= 30000 s Var(Payment) = (1.5) 2 Var(X), Var(X)= 8000 2 5 Are we understanding this stuff ? A contractor estimates the probabilities for the number of days required to complete a certain type of job as follows: Time (Days) 1 2 3 4 5 6 Probability .05 .20 .35 .30 .10 What is the probability that a project will take less than 3 days to complete ? Time (Days) 1 2 3 4 5 Probability .05 .20 .35 .30 .10 Here is the table again: Find the expected time and variance of time required to complete a project. ( ) 1(.05) 2(.20) 3(.35) 4(.30) 5(.10) 3.2 E X = + + + + = 7 σ = = + + + + = 2 2 2 2 2 2 ( ) (1 3.2) (.05) (2 3.2) (.2) (3 3.2) (0.35) (4 3.2) (0.3) (5 3.2) (0.1) 1.06 X Var X The contractor’s project cost is made up of two partsa fixed cost of $20000 plus $2000 for each day taken to complete the task. Find the mean and standard deviation of total project cost. 20000 2000* Cost X = + 8 2 ( ) 20000 2000 ( ) 20000 2000(3.2) 26400 ( ) (2000) ( ) (4000000)(1.06) E Cost E X Var Cost Var X = + = + = = = Why the a+bX rule? s We could have done the following 9 b That is, just create a new probability table, and find the new mean and variance from this new probability table. b But what if you don’t even have the original probability table? Example: Buffalo Eyeballs s On the new show “Buffalo Eyeball Factor”, each contestant is required to eat a randomly determined number of buffalo eyeballs; call this random variable B. s The expected value of B is 5, and the standard deviation is 4. s Each contestant gets $100 just for appearing on the Buffalo Eyeball show, and an additional $20 for each buffalo eyeball they eat....
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.
 Fall '11
 MichaelParzen
 Variance

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