Download Linear Functions of a Random Variable

Engineering Statistics ECIV 2305
Section 2.6
Combinations & Functions of Random Variables
Linear Functions of a Random Variable

If X is a random variable and Y is a linear function of
the random variable X, where
Y = aX +b
(a & b are numbers)
then it is a general rule that:
E(Y) = aE(X) +b
and Var(Y) = a2 Var(X)
(how?)
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Linear Functions of a Random Variable

An important application of this result will be used in
chapter 5, which concerns the “standardization” of a
random variable X to have a zero mean and a unit
variance. The new “standardized” random variable
will be:
X  1
 
Y

X   

 

If you apply the previous linear function rule, then
1
 
E (Y )  a.E ( X )  b  .      zero

 
2
1 2
Var (Y )  a .Var ( X )    .  1
 
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…Linear Functions of a Random Variable

In order to find the pdf of the new transfromed
random variable Y, one should do the following:



Start with the pdf of the random variable X
Find the cdf of the random variable X , i.e. FX(x)
Convert from the cdf of X to the cdf of Y


i.e. from FX(x) to FY(y)
Differentiate the cdf of Y to get the pdf of Y
The new part here is how to convert from the cdf of X
to the cdf of Y
FY  y   PY  y   PaX  b  y   PaX  y  b
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…Linear Functions of a Random Variable
FY  y   PY  y   PaX  b  y   PaX  y  b
 If a  0, this gives
y b

 y b
FY  y   P X 
  FX 

a 

 a 
 If a  0, this gives
y b 

 y b
FY  y   P X 
  1  FX 

a 

 a 
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Example: Test Score Standardization (page 142)
Suppose that the raw scores X from a particular testing
procedure are distributed between -5 and 20 with an
expected value of 10 and a variance of 7.
a) It is required to standardize the scores so that they
lie between 0 and 100 using a linear transformation.
b) Find the expected value and the standard deviation
of the new standardized scores.
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…example: Test Score Standardization (page 142)
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Example: Chemical Reaction Temperatures (page 142)
The temperature X in degrees Fahrenheit of a particular
chemical reaction is known to be distributed between
220o and 280o with a probability density function of
x  190
f X ( x) 
3600
With an expectation of E(X) = 255o and a standard
deviation of σX = 16.58o.
Suppose that the chemist wishes to convert the
temperatures to degrees Centigrade. Find the pdf of
the temperature in degrees Centigrade along with the
new mean and standard deviation in degrees
Centigrade
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… Example: Chemical Reaction Temperatures (page 142)
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… Example: Chemical Reaction Temperatures (page 142)
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Linear Combinations of Random Variables
If X1, X2, …….. , Xn is a sequence of random variables
and a1, a2, ….., an and b are constants, and Y is a linear
combination in the following form
Y = a1X1 + a2X2 + …….. + anXn + b
then

E(Y) = a1E(X1) + a2E(X2) + …….. + anE(Xn)+ b
and
Var(Y) = (a1)2Var(X1)+ (a2)2Var(X2)+…+ (an)2Var(Xn)
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Example:
page 146
Suppose that X1 and X2 are two independent random
variables and both of them have an expectation of μ and
a variance of σ2 .
and suppose that Y1 and Y2 are two random variable
for which
Y1 = X1 + X2
Y2 = X1 – X2
Find the expectations and variances of Y1 and Y2.

E(Y1)= E(X1)+ E(X2) = 2μ and E(Y2)= E(X1) – E(X2) = zero
Var(Y1)= (1)2Var(X1)+ (1)2Var(X2) = 2σ2 and
Var(Y2)= (1)2Var(X1)+ (-1)2Var(X2) = 2σ2
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Conclusion from the previous example:

Adding or subtracting independent random variables
increases variability
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Averaging Independent Random Variables

Suppose that X1 , X2, ……, Xn is a sequence of
independent random variables each with an expectation
μ and a variance of σ2 , and with an average of X
X 1  X 2  ........  X n
X
n
then
E( X )  
and Var ( X ) 
2
n
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