Distribution functions, Moments, Moment generating functions in the Multivariate case

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Distribution functions, Moments, Moment
generating functions in the Multivariate case
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The distribution function F(x)

• This is defined for any random variable, X.

F(x) PX x
Properties

1. F(-8) 0 and F(8) 1.

• F(x) is non-decreasing
• (i. e. if x1 lt x2 then F(x1) F(x2) )

1. F(b) F(a) Pa lt X b.

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1. Discrete Random Variables

F(x)
p(x)
F(x) is a non-decreasing step function with
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1. Continuous Random Variables Variables

F(x) is a non-decreasing continuous function with
To find the probability density function, f(x),
one first finds F(x) then
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The joint distribution function F(x1, x2, , xk)

• is defined for k random variables, X1, X2, , Xk.

F(x1, x2, , xk) P X1 x1, X2 x2 , , Xk
xk
x2
for k 2
(x1, x2)
x1
F(x1, x2) P X1 x1, X2 x2
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Properties

1. F(x1 , -8) F(-8 , x2) F(-8 , -8) 0
2. F(x1 , 8) P X1 x1, X2 8 P X1 x1
   F1 (x1)

the marginal cumulative distribution function
of X1
F(8, x2) P X1 8, X2 x2 P X2 x2 F2
(x2)
the marginal cumulative distribution function
of X2
F(8, 8) P X1 8, X2 8 1
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1. F(x1, x2 ) is non-decreasing in both the x1
   direction and the x2 direction.

• i.e. if a1 lt b1 if a2 lt b2 then
• F(a1, x2) F(b1 , x2)
• F(x1, a2) F(x1 , b2)
• F( a1, a2) F(b1 , b2)

x2
(b1, b2)
(a1, b2)
x1
(a1, a2)
(b1, a2)
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1. Pa lt X1 b, c lt X2 d

F(b,d) F(a,d) F(b,c) F(a,c).
x2
(b, d)
(a, d)
x1
(a, c)
(b, c)
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1. Discrete Random Variables

x2
(x1, x2)
x1
F(x1, x2) is a step surface
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1. Continuous Random Variables

x2
(x1, x2)
x1
F(x1, x2) is a surface
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Multivariate Moments

• Non-central and Central

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• Definition
• Let X1 and X2 be a jointly distirbuted random
  variables (discrete or continuous), then for any
  pair of positive integers (k1, k2) the joint
  moment of (X1, X2) of order (k1, k2) is defined
  to be

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• Definition
• Let X1 and X2 be a jointly distirbuted random
  variables (discrete or continuous), then for any
  pair of positive integers (k1, k2) the joint
  central moment of (X1, X2) of order (k1, k2) is
  defined to be

where m1 E X1 and m2 E X2
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• Note

the covariance of X1 and X2.
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Multivariate Moment Generating functions
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• Recall

The moment generating function
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• Definition
• Let X1, X2, Xk be a jointly distributed random
  variables (discrete or continuous), then the
  joint moment generating function is defined to be

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• Definition
• Let X1, X2, Xk be a jointly distributed random
  variables (discrete or continuous), then the
  joint moment generating function is defined to be

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• Power Series expansion the joint moment
  generating function (k 2)

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The Central Limit theorem

• revisited

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The Central Limit theorem
If x1, x2, , xn is a sample from a distribution
with mean m, and standard deviations s, then if n
is large

has a normal distribution with mean
and variance
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The Central Limit theorem illustrated
If x1, x2 are independent from the uniform
distirbution from 0 to 1. Find the distribution
of

let
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Now

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Now

The density of
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n 1
1
0
n 2
1
0
n 3
1
0