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Unformatted text preview: I Chapter 5  Expected Value 5.3 The Standard Deviation of a Random Variable The Distribution Formula for the Standard Deviation of a Random Variable
The theoretical variance of X, denoted either by Var(X) or a}, , is computed by 0X =1_(X ,uX)2 p(X1)+(X2 _#X)2 p(X2)+"'+(X/( #X.)Z p(Xk) = EMT/[XV p(X)'
The (theoretical) standard deviation of X, denoted either by SD(X) or 0X , is the square
root of the variance: 50<X>=ox =JE=\/Z<X—u.)2p(x) Example 10: Roll a Die q'olIg {5 Using simulation, ﬁnd the expected value for the sum whenten—diee—are rolled andthe mbersen—the—dieeareadded. Then, using the formulas above, calculate the variance
and the standard deviation of X. :=(l'3537‘%(>+ (“L 3536g>+‘~‘+(c 353G?) _ M(Z><C'Z .25) 4 (ll331 +C~0 531+(oI—33L+<[.53 +<z5))
, (2: 3/6 25+ 225 +0. 5 +0.25+2.25+4.25) (.3075): m
0‘x = V2.9“: a: Theoretical 68% 95% 99. 7% Rule
Suppose the distribution of X is roughly bell—shaped. Then:
1. There is about a 68% chance that X will fall in the range W i ox. 2. There is about a 95% chance that X will fall in the range llx i 20x. 3. There is about a 99.7% chance that X will fall in the range ux i 36x. . YE 5'.
Example 11: Rolla Die A is this be“ shaped 7. Ear Mulls, 14,.“  35,
Using the answers from Example 10, answer the following questions. 5;... Q g ,1
a) There is about a 68% chance that X will fall in what range? ‘ a 5 a :1 =
b) There is about a 95% chance that X ' II in what ran e?
55: 2' BH =]2_H.1 ~ H5. 8} c) There is about a 99.7% chance that X will fall in what range? 35:3»SH = {8.8 — 5L7. 65. Chapter 5 — Expected Value Box Model Formulas for Box Means, Variance, and Standard Deviation
Consider a box with N numbered balls. Let X be the result of one random draw. Then
the expected .value of X equals the box mean: #X :Iubox =7 where we sum over all the balls in the box. The variance of X equals the box
variance: ’ Z X_Iu0X2
0X2020x22( N b L! I! U The standard deviation of X equals the box standard deviation, which is the square
root of the box variance. __ _ 2
0X — abox _ V Ubox ‘ Box Model Formulas where Y is a Sum of 11 Random Draws from a Box
If Y is the sum of the numbers on n randomly selected balls from a box, drawn with
replacement between draws, then a)’ = J; ' abox If Y is the sum of the numbers on n randomly selected balls from a box, drawn without
replacement between draws, then 0Y:‘/77_'0box’ /v—1’ where N = number of balls inthe box. If we draw without replacement, then the
sample size n cannot be greater than the total number of balls N. Here [N — n. N — 1
is called the ﬁnite population (adjustment) factor to the withreplacement formula
for (W due to the ﬁnite “population size” N when sampling without replacement. 66 Chapter 5 — Expected Value 1? Example 12: Spinner, revisited W
In example 5 we calculated the expected value of a spin for the spinner shown below. Set up a box model for the spinner and calculate the variance and standard deviationﬁif
we spin 50 times. ' A“ {hast"'1
[I 12235 L15 E[on¢59.\n1=z.75
n=50 w/replacemm‘b CCXamPlcE)
50:: Sum
Mean : 50./Moneo\rnw : 50' (2'76 ‘2 Z  z _ . . : .
0—34!“ _ 50.0Medm‘g  50 lH375 7! 875 6.... = W50 « 2......: iso  “"375 Example 13: Board Game, revisited 
In Example 6, we calculated the expected value for a board game where players roll a die and then advance the number of spaces shoWn on the die. After 50 rolls, a player should expect to advance around I 7 5 spaces give or take (‘2 .0 7 7
. I :— 3, '5 : ,
or so spaces ,{1 . 6;!1: (on I. 70 5 on: (ad ”SUM : Eo'ﬂoner‘ou : 50'35 :
6—511!” : [lg—glojnefl’olll : Va: ‘l708 : After 100 rolls, a player should expect to advance around 3 5 0 spaces give or take (7. O 8 or so spaces. H‘suM ; [ODIMO": Fol/l : [00' 3‘5 :@ sum“ too aroma“ IO 1'7 8 © 57 ' Chapter 5 — Expected Value Example 14: Sum of Balls :75
Let X be the sum of the numbers on E balls drawn at random with replacement from “mooooeo a) What is the smallest X can be?
‘5  17 I?) = 87 5 b) What is the largest X can be? c) Most likely, X will be around 30 6 Z . 5 , give or take l IZ ‘7 7 or so. ”Onedram: 5M0M(o\\ofaA\‘<. I: 531:5): ‘7 5
0—0”? All“: 5 6;“: rolldfqali'é : 6’ "708: 85% W
Méum : '76' Moﬂedra“) :' 1.75 ' l7.;_) I @ 6—50.11“ 1 M ' ﬁncdﬁaw : I75 . 95% ”'5' When there are only two different values on the tickets in your box model, there is a
shortcut formula that you can use to calculate the standard deviation. If your box model is as follows, a b
I then the standard deviation of the box is ab
(a + b)2 ' o=X—y 68 ...
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 Spring '09
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