16 sequences

Hadley Wickham

Stat310Sequences of rvs

Wednesday, 17 March 2010

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


Assessment

Test model answers online tonight (hopefully)

Usual help session tonight 4-5pm.


1. Sequences

2. Limits

3. Chebyshev’s theorem

4. The law of large numbers

5. The central limit theorem


1 variable: X

2 variables: X, Y

...

n variables: X1, X2, X3, ..., Xn

Sequences


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.


iid = independent & identically distributed


Your turn

Xi are iid N(0, 2).

What is E(X30)? What is Var(X2001)?

What is Cor(X10, X11)? Cor(X1, X1000)?


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?


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


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

.


Chebyshev

P (|X − µ| < Kσ) ≥ 1− 1K2

P (|X − µ| > Kσ) ≤ 1K2

For K > 0

No limit - but a

good starting point


Your turn

How can you put this in words?

P (|X − µ| > Kσ) ≤ 1K2


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)


K

1K2

0.0

0.2

0.4

0.6

0.8

1.0

2 4 6 8 10

(For K > 1)


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.


x

value

0.0

0.2

0.4

0.6

0.8

1.0

2 4 6 8 10

variablechebynorm


LLNLaw of large numbers

X1, X2, ..., Xn iid.

There are five ways to write the result.

X̄n =n�

i

Xi


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


CLT

Central limit theorem.

The distribution of a mean is normal when gets big.


Approximation

This implies that if n is big then ...


Reading

Section 4.1

Focus on the general ideas and the defintions


16 sequences

Technology

xn iid

n variables

turn xi

nne xi

x2 variables

closer tothe true mean

2k1p x

iid n0