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Hadley Wickham
Stat310Sequences of rvs
Wednesday, 17 March 2010
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Major’s day
2:30-4:30pm Today
Oshman Engineering Design Kitchen
Come along and talk to me (or Rudy Guerra) if you’re interested in becoming a stat major
Wednesday, 17 March 2010
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Assessment
Test model answers online tonight (hopefully)
Usual help session tonight 4-5pm.
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1. Sequences
2. Limits
3. Chebyshev’s theorem
4. The law of large numbers
5. The central limit theorem
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1 variable: X
2 variables: X, Y
...
n variables: X1, X2, X3, ..., Xn
Sequences
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SequencesXi ~ Normal(μi, σi)
Xi ~ Normal(μ, σi)
Xi ~ Normal(μi, σ)
Xi ~ Normal(μ, σ)
Almost always assume that the Xi’s are independent. In the last case they are also identically distributed.
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iid = independent & identically distributed
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Your turn
Xi are iid N(0, 2).
What is E(X30)? What is Var(X2001)?
What is Cor(X10, X11)? Cor(X1, X1000)?
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E(n�
i
Xi) =n�
i
E(Xi)
V ar(n�
i
aiXi) =n�
i
a2i V ar(Xi)
If what is true?
E(n�
i
Xi) =n�
i
E(Xi)If what is true?
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Limits
Typically will define some function of n random variables, e.g.
What happens to when n → ∞?
Why? Because often it will converge, and we can use this to approximate results for any large n.
X̄n
X̄n
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New notation
If xn → 0, and n is big, we can say xn ≈ 0.
If Xn → Z, Z ~ N(0, 1), and n is big, we can say Xn ~ N(0,1).
Read as approximately distributed.
Other ways to write it
.
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Chebyshev
P (|X − µ| < Kσ) ≥ 1− 1K2
P (|X − µ| > Kσ) ≤ 1K2
For K > 0
No limit - but a
good starting point
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Your turn
How can you put this in words?
P (|X − µ| > Kσ) ≤ 1K2
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K
1K2
20
40
60
80
0 2 4 6 8 10
The probability of being more than K standard deviations away from the mean is less than one over K squared.
(For K > 0)
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K
1K2
0.0
0.2
0.4
0.6
0.8
1.0
2 4 6 8 10
(For K > 1)
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Your turn
How does this compare to the normal distribution? Compare the probability of being less than 1, 2 and 3 standard deviations away from the mean given by Chebychev and what we know about the normal.
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x
value
0.0
0.2
0.4
0.6
0.8
1.0
2 4 6 8 10
variablechebynorm
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LLNLaw of large numbers
X1, X2, ..., Xn iid.
There are five ways to write the result.
X̄n =n�
i
Xi
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What does it mean?
As we collect more and more data, the sample mean gets closer and closer to the true mean.
Not that surprising!
But note that we didn’t make any assumptions about the distributions
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CLT
Central limit theorem.
The distribution of a mean is normal when gets big.
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Approximation
This implies that if n is big then ...
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Reading
Section 4.1
Focus on the general ideas and the defintions
Wednesday, 17 March 2010