Linear Combination of Two Random Variables
• Let X1 and X2 be random variables with
E  X 1   1 , E  X 2    2 , var  X 1    12 , var  X 2    22
and
Cov X 1 , X 2    12
• Let Y = aX1 + bX2. Then,…
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Linear Combination of Independent R.V
• Let X1, X2,…, Xn be independent random variables with
EX i   i
•
and
var  X i    i2
Let Y = a1X1 + a2X2 + ∙∙∙+ anXn. Then,…
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Linear Combination of Normal R.V
• Let X1, X2,…, Xn be independent normally distributed random
variables with
EX i   i
and
var  X i    i2
• Let Y = a1X1 + a2X2 + ∙∙∙+ anXn. Then,…
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Examples
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The Chi Square distribution
• If Z ~ N(0,1) then, X = Z2 has a Chi-Square distribution with
parameter 1, i.e., X ~  21 .
• Can proof this using change of variable theorem for univariate
random variables.
• The moment generating function of X is
1/ 2
 1 
m X t   

 1  2t 
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Important Facts
2
2
2
• If X 1 ~  v1  , X 2 ~  v2  , , X k ~  vk  , all independent then,
k
T   X i ~  2k v
i 1
1 i
• If Z1, Z2,…Zn are i.i.d N(0,1) random variables. Then,
Y  X 12  X 22    X n2 ~  2n 
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Claim
• Suppose X1, X2,…Xn are i.i.d normal random variables with mean μ
and variance σ2. Then, Z i  X i   are independent standard normal

variables, where i = 1, 2, …, n and
 Xi   
2
Z

~





n 
i
 
i 1
i 1 
n
2
n
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Important facts
• Suppose X1, X2,…Xn are i.i.d normal random variables with mean μ
and variance σ2. Then,
1
2
 X
n
i 1
 X  ~  2n 1
2
i
Proof:…
• This implies that…
n  1S 2

2
~  2n 1
• Further, it can be shown that X and s2 are independent.
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