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Lecture 4 Rescaling, Sum and difference of random variables: simple algebra for mean and standard deviation (X+Y)2=X2 + Y2 + 2 XY E (X+Y)2 = EX2 + EY2 + 2 EXY Var (X+Y) = Var (X) + Var (Y) if independence Demonstrate with Box model (computer simulation) Two boxes : BOX A ; BOX B Each containing “infinitely” many tickets with numeric values (so that we don’t have to worry about the estimation problem now; use n)

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Change of scale Inch to centimeter: cm= inch times 2.54 pound to kilogram: kg=lb times 2.2 Fahrenheit to Celsius oC= ( oF-32)/1.8 Y= X+a E Y = E X + a SD (Y) = SD (X) ; SD(a) =0 Y= c X E Y = c E X SD (Y)= |c| SD(X); Var (Y)= c2Var (X) Y=cX + a EY= c E X + a SD (Y) =| c| SD (X); Var (Y)= c2 Var(X) Var X= E (X-)2= E X2 - (EX)2 (where = E X)

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BOX A

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Two Boxes A and B ; independence

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E (X+ Y) = E X + E Y; always holds E ( X Y) = ( E X ) ( EY) ; holds under independence assumption (show this! Next) Without independence assumption E(XY) is in general not equal to EX times EY ; it holds under a weaker form of independence called “uncorrelatedness” (to be discussed )

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Combination Var (a X + b Y) = a2 Var X + b2 Var Y if X and Y are independent Var (X-Y) = Var X + Var Y Application : average of two independent measurement is more accurate than one measurement : a 50% reduction in variance Application : difference for normal distribution

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Example Phone call charge : 40 cents per minute plus a fixed connection fee of 50 cents Length of a call is random with mean 2.5 minutes and a standard deviation of 1 minute. What is the mean and standard deviation of the distribution of phone call charges ? What is the probability that a phone call costs more than 2 dollars? What is the probability that two independent phone calls in total cost more than 4 dollars? What is the probability that the second phone call costs more than the first one by least 1 dollar?

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Example Stock A and Stock B Current price : both the same, $10 per share Predicted performance a week later: same Both following a normal distribution with Mean $10.0 and SD $1.0 You have twenty dollars to invest Option 1 : buy 2 shares of A portfolio mean=?, SD=? Option 2 : buy one share of A and one share of B Which one is better? Why?

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Better? In what sense? What is the prob that portfolio value will be higher than 22 ? What is the prob that portfolio value will be lower than 18? What is the prob that portfolio value will be between18 and 22? (draw the distribution and compare)