Transcript Document 7920980

730 Lecture 2
Today’s lecture:
Inequalities
Convergence
Taylor Series
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Inequalities
Chebychev :

P(| X   |  )  2

2
Jensen :
g[ E ( X )]  E[ g ( X )]
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Chebychev
Area Area
2/2
x  0,1,..., n
  +
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Chebychev: proof


( x   )2
 
f ( x)dx
2
2



2
 


 

(x  )

2
2
f ( x)dx +


 +
( x   )2

2
f ( x)dx

 f ( x)dx + +f ( x)dx
 P[ X     ] + P[ X   +  ]
 P[| X   |  ]
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Jensen
g(x)
g() + g’()(x-)
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
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Example : sample standard
deviation
• Graph of y=x:
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
2
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Std dev (cont)
• Slope at x=2 is 1/2
• Tangent at point (,2) lies above curve
• THUS
s2  2
s  +
2
Taking expectatio ns
E (s 2   2 )
E(s)   +

2
Sample sd BIASSED DOWN!
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Convergence
• Convergence in distribution
P( X n  x )  P ( X  x )
• Main use: approximation Eg
CLT
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Examples
• tn converges to normal
• Binomial converges to Poisson
• Gamma converges to normal
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Central limit Theorem
(CLT)
 n(X  )

P
 x    ( x)



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Example
Convergence of binomial to Normal:
Express Xn as
n
X n   Yi where Y1 ,..., Yn are independen t with
i 1
Yi  1 with probabilit y p,
 0 with probabilit y 1  p.
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Example (cont)
Then result is just CLT, as the Y’s have
mean p and variance p(1-p).
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Example
• Convergence of binomial to Poisson:
• Proved by showing convergence of the
prob. function since
[ x]
Fn ( x)   pn ( j )
j 0
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Example 5 (cont)
npn n
n n  1 n  y + 1 (npn ) y
y
.
...
.
.(1 
) .(1  pn)
n n
n
y!
n
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Convergence (2)
Convergence in probability: Xn
converges to X “in probability” if
P(| X n  X |  )  0
Eg proportion of heads in n
tosses 
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Example
Xn=proportion of heads in n tosses
E(Xn)=p=0.5, Var(Xn)=p(1-p)/n=1/4n
By Chebychev,
P(|Xn-0.5|>)  Var(Xn)/2 =1/(4n 2)  0
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More examples
• Sample mean converges in probability to
population mean as sample size gets
larger
(Weak law of large numbers)
• Sample variance converges to population
variance
• “Consistent” estimators
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Example: sample variance
n
1
2
s2 
(
X

X
)

i
n  1 i 1
n
1

{ ( X i   ) 2  n( X   ) 2}
n  1 i 1
n
1

{ Yi  n( X   ) 2}
n  1 i 1
n
n

Y
( X   )2
n 1
n 1
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Sample variance (cont)
Y  E (Yi )  E ( X i   )   ,
2
2
X    0, so ( X   )  0  0
2
2
Thus
s  1   1 0   .
2
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Example
X1,…Xn Uniform
Mn=max (X1,…Xn )
P(Mn  x) =xn (why?)
P(|Mn- 1|>)  P(Mn<1- ) = (1- )n  0
Thus Mn converges to 1 in probability
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