Download 6.2 Transforming and Combining Random Variables

6.2 Transforming and Combining Random Variables
LINEAR TRANSFORMATIONS
Recall:
1. Adding (or Subtracting) a constant: Adding the same number a (either positive, zero, or negative) to each observation:
a) Adds a to measures of center and location
(mean, median, quartiles and percentiles)
b) Does not change shape or measure of spread
(range, IQR, standard deviation)
2. Multiplying (or Dividing) each observation by a constant b
(positive, negative, or zero):
a) Multiplies(divides) measures of center and location by b
(mean, median, quartiles and percentiles)
b) Multiplies (divides) measures of spread by |b|.
(range, IQR, standard deviation)
c) Does not change the shape of the distribution.
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Effect on a Random Variable of Multiplying(or Dividing) by a Constant
Multiplying (or Dividing) each value of a random variable by a number b:
1. Multiplies (divides) measures of center and location by b. (mean, median, quartiles and percentiles)
2. Multiplies (divided) measures of spread by |b|.
(range, IQR, standard deviation)
3. Does not change the shape of the distribution.
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Example: Pete's Jeep Tours start p358
No. of pass xi
2
Prob pi
.15 .25 .35
3
4
5
6
.20 .05
μx = (2)(.15) + (3)(.25) + (4)(.35) +(5)(.20) + (6)(.05) = 3.75
σx2 = (2­3.75)2(.15) +(3­3.75)2(.25)+....+(6­3.75)2(.05) = 1.1875
so standard deviation of X is σx = √1.1875 = 1.090
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Pete's Jeep P259
total collected 300 450 600
ci
750 900
Prob pi
.20 .05
.15 .25 .35
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Effect on a Random Variable of Adding (Subtracting) a Constant
Adding the same number a to each value of a random variable
1. Adds a to measures of center and location.
(mean, median, quartiles and percentiles)
2. Does not change shape or measures of spread.
(range, IQR, standard deviation)
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Pete's Jeep Tours p360
total collected 300 450 600
ci
750 900
Prob pi
.20 .05
.15 .25 .35
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If we do both add/subtract and mult./divide then creates a Linear Transformation of the random variable X.
Effects of a Linear Transformation on the Mean and Standard Deviation
If Y = a + bX is a linear transformation of the random variable X, then 1. The probability distribution of Y has the same shape as the probability distribution of X.
2. μx = a + bμx
3. σy = |b|σx (since b could be a negative number)
Whether we're dealing with data or random variables, the effects of a linear transformation are the same. Note that these results apply to both discrete and continuous random variables.
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the Baby and the Bathwater p363
F = 9/5C +32
a)
μy = 32 +9/5μx = 32 + 9/5(34) = 93.2 Fahrenheit
σy = 9/5σx = 9/5(2) = 3.6 degrees F
b) z = (90 ­ 93.2)/3.6 = ­.89 and z = (100 ­ 93.2)/3.6 = 1.89
Then the P(­.89≤ z ≤ 1.89) = .9706 ­ .1867 = .7839
There's about a 78% chance that the water temp. meets the recommendation on a randomly selected day.
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Example: Put a Lid on It. p. 364
Sums of Random Variables p364
No. of pass. 2
xi
Prob .15
pi
No pass 2
yi
Prob pi .3
3
4
5
6
.25
.35
.20
.05
3
4
5
.4
.2
.1
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COMBINING RANDOM VARIABLES
MEAN OF THE SUM OF RANDOM VARIABLES
For any two random variables X and Y, if T =X + Y, then the expected value of T is
E(T) = μT = μX + μY
In general, the mean of the sum of several random variables is the sum of their means.
Ex: Let X1, X2, X3 be random variables
μ1 =1
μ2 =2
μ3 =­1
What's the mean of X1 ­ 2X2 + X3 ?
So we want to find E(X1 ­ 2X2 + X3) = μ1­2μ2 + μ3
= 1 ­ 2(2) + ­1 = 1­4­1 = ­4
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Example: Let X1, X2.......Xn be independent random variables all with the same mean.
What is the mean of X?
E(X) = E(1/n(X1+ X2+.......+Xn))
= 1/n(μ+μ+μ +.......+μ) = 1/n(nμ) = μ
So expected value of X is μ
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Definition: Independent Random Variables
If knowing whether any event involving X alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then X and Y are independent random variables.
Example: p366 Pete's Jeep Tours and Erin's Adventures
Value 4
of ti
5
6
7
8
prob pi .045 .135 .235 .265 .190
9
10
.095 .03
11
.005
The Mean of T = and recall the mean of t = mean of x + mean of y
Variance of T = standard deviation of T = 12
VARIANCE OF THE SUM OF INDEPENDENT RANDOM VARIABLES
For any two independent random variables X and Y, if T = X +Y, then the variance of T is
σ2T = σ2X +σ2Y
In general, the variance of the sum of several independent random variables is the sum of their variances.
Can add variances only if they are independent!
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Example: Let X1, X2.......Xn be random variables. Find Var(X1 ­ 2X2 + X3) = σ12+(­2)2σ22 + σ32
= 1 + 4(3) +5 = 18
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Example: Let X1, X2.......Xn be independent random variables all with the same variance.
What is the variance of the sample mean X?
Var(X) = Var(1/n(X1+ X2+.......+Xn))
= 1/n2(σ2+σ2+σ2 +.......+σ2) = 1/n2(nσ2) = 1/n(σ2)
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Example: SAT Scoresp.368
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Example: Pete's Jeep and Erin's Tours P368
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Think About It ­ X1 +X2 is not the same as 2X
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MEAN OF THE DIFFERENCE OF RANDOM VARIABLES
For any two random variables X and Y, if D = X ­ Y, then the expected value of D is
E(D) = μD = μX ­ μY
In general, the mean of the difference of several random variables is the difference of their means.
Order of subtraction is important!
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VARIANCE OF THE DIFFERENCE OF RANDOM VARIABLES
For any two independent random variables X and Y, if D = X ­ Y, then the variance of D is
σ2D = σ2X ­ σ2Y
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Example: Pete's Jeep and Erin's Tours p372
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COMBINING NORMAL RANDOM VARIABLES
Example: A computer simulation p373
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Example: Give me some sugar p374
Using normalcdf(8.5,9,8.68,0,16) or Normal curve applet yields P(8.5≤ T ≤ 9) = .8470
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Put a Lid on It! P375
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Technology Corner : Simulating with randNorm
AP Exam Tip: If you have trouble solving problems involving sums and differences of Normal random variables with the algebraic methods shown earlier, use the simulation strategy from the Tech corner to earn some credit on the AP Exam.
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Homework
p356 and 378
p379
27­30, 37, 39­41, 43,45
49,51, 57­59, 63
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