This document contains notes from a statistics course. It discusses sequences of random variables, limits, Chebyshev's theorem, the law of large numbers, and the central limit theorem. The notes provide definitions and examples of key concepts like independent and identically distributed random variables, convergence in probability, and how the sample mean approximates the population mean as the sample size increases. Students are prompted with practice problems applying the concepts.

1. Stat310 Sequences of rvs
Hadley Wickham
Wednesday, 17 March 2010

2. 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

3. Assessment
Test model answers online tonight
(hopefully)
Usual help session tonight 4-5pm.
Wednesday, 17 March 2010

4. 1. Sequences
2. Limits
3. Chebyshev’s theorem
4. The law of large numbers
5. The central limit theorem
Wednesday, 17 March 2010

5. Sequences
1 variable: X
2 variables: X, Y
...
n variables: X1, X2, X3, ..., Xn
Wednesday, 17 March 2010

6. Sequences
Xi ~ 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.
Wednesday, 17 March 2010

7. iid = independent &
identically distributed
Wednesday, 17 March 2010

8. Your turn
Xi are iid N(0, 2).
What is E(X30)? What is Var(X2001)?
What is Cor(X10, X11)? Cor(X1, X1000)?
Wednesday, 17 March 2010

9. n
n
E( Xi ) = E(Xi )
i i
n
n
V ar( ai Xi ) = 2
ai V ar(Xi )
i i
If what is true?
n
n
E( Xi ) = E(Xi )
i i If what is true?
Wednesday, 17 March 2010

10. Limits
Typically will define some function of n
¯
random variables, e.g. Xn
¯
What happens to Xn when n → ∞?
Why? Because often it will converge, and
we can use this to approximate results for
any large n.
Wednesday, 17 March 2010

11. 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
Wednesday, 17 March 2010

12. N
go
o
od
lim art
Chebyshev
it ing
st
-b p
ut oin
a t
1
P (|X − µ| Kσ) ≥ 1 − 2
K
1
P (|X − µ| Kσ) ≤ 2
K
For K 0
Wednesday, 17 March 2010

13. Your turn
How can you put this in words?
1
P (|X − µ| Kσ) ≤ 2
K
Wednesday, 17 March 2010

14. The probability of being more
than K standard deviations
80 away from the mean is less
than one over K squared.
60
(For K 0)
1 K2
40
20
0 2 4 6 8 10
K
Wednesday, 17 March 2010

15. (For K 1)
1.0
0.8
0.6
1 K2
0.4
0.2
0.0
2 4 6 8 10
K
Wednesday, 17 March 2010

16. 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.
Wednesday, 17 March 2010

17. 1.0
0.8
0.6
variable
value
cheby
norm
0.4
0.2
0.0
2 4 6 8 10
x
Wednesday, 17 March 2010

18. LLN
Law of large numbers
X1, X2, ..., Xn iid.
n
¯
Xn = Xi
i
There are five ways to write the result.
Wednesday, 17 March 2010

19. 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
Wednesday, 17 March 2010

20. CLT
Central limit theorem.
The distribution of a mean is normal when
gets big.
Wednesday, 17 March 2010

21. Approximation
This implies that if n is big then ...
Wednesday, 17 March 2010

22. Reading
Section 4.1
Focus on the general ideas and the
defintions
Wednesday, 17 March 2010