1. Stat310
Sums of independent rv’s
Hadley Wickham
Tuesday, 10 March 2009

2. http://xkcd.com/552/
Tuesday, 10 March 2009

3. 1. Exam
2. Recap
3. Motivation - connection to data
4. Expectation, variance, mgf’s
5. Convergence in probability
Tuesday, 10 March 2009

4. Exam
How did it go? Too long? Too short?
Sorry about question 1
Draft model answers online
Will be back to you next week
Tuesday, 10 March 2009

5. Tuesday, 10 March 2009

6. Recap
What is a random experiment?
What does it mean for two random
variables to be independent?
What is E(X + Y)?
What is Var(X + Y)?
What is the definition of the mgf?
Tuesday, 10 March 2009

7. Random experiment
An observation that is uncertain:
we don’t know ahead of time what the
answer will be (pretty common!)
Ideally we, want the experiment to be
repeatable under exactly the same initial
conditions (pretty rare!)
Tuesday, 10 March 2009

8. Why?
Want to be able to work with (many) more
variables than just two.
Typically need to make some
assumptions to make this easier.
Turns out many of the things we are
interested in can be expressed as sums,
and we can design experiments to ensure
independence.
Tuesday, 10 March 2009

9. Why?
Most often these random variables will
arise from repeated experiments. Each
random variable is a result from a repetition.
Want to be able to make inferences about
the distribution from the samples that we
see.
Will touch on this today, and cover much
more deeply in two weeks.
Tuesday, 10 March 2009

10. Mutual independence
If X1, X2, X3, …, and Xn are mutually
independent, what do you think the joint
pdf will look like?
f(x1, x2, …, xn) = ?
Tuesday, 10 March 2009

11. f (x1 , x2 , . . . , xn ) = f (x1 )f (x2 ) · · · f (xn )
f (x1 , x2 , . . . , xn ) = f1 (x1 )f2 (x2 ) · · · fn (xn )
n
= fi (xi )
i=1
Tuesday, 10 March 2009

12. Note
If X and Y are independent, then so are:
X2 and Y2
1/X and eY
u(X) and v(Y) - any functions that only
involve one variable
Tuesday, 10 March 2009

13. Definition
iid = independent identically distributed
e.g. X1, X2, …, Xn iid implies that
f (xi )
f(x1, x2, …, xn) =
And the f’s are all the same!
Tuesday, 10 March 2009

14. Mean & Variance
Let X1, X2, …, Xn be independent random
variables. Then:
Xi = E(Xi )
E
Xi = V ar(Xi )
V ar
Tuesday, 10 March 2009

15. Y = a1 X1 + a2 X2 + · · · + an Xn
What are E(Y) and Var(Y) ?
Tuesday, 10 March 2009

16. Y = a1 X1 + a2 X2 + · · · + an Xn
What are E(Y) and Var(Y) ?
E(Y ) = ai E(Xi )
Tuesday, 10 March 2009

17. Y = a1 X1 + a2 X2 + · · · + an Xn
What are E(Y) and Var(Y) ?
E(Y ) = ai E(Xi )
V ar(Y ) = ar(Xi )
2
ai V
Tuesday, 10 March 2009

18. Your turn
Let X1, X2, …, Xn be iid random variables.
Xi
¯=
X
n
What are the mean and variance?
Tuesday, 10 March 2009

19. 1
¯=
X Xi
n
¯ = E(X)
E(X)
V ar(X)
¯=
V ar(X)
n
Tuesday, 10 March 2009

20. Mgf
So that’s great if all we want to know is
the mean and variance. What if we want
to know the actual distribution?
This is where we use the second property
of the mgf - if the mgfs of two random
variables are equal, then the distributions
(pdfs) are equal
Tuesday, 10 March 2009

21. Your turn
If X and Y are independent, what is the
mgf of X + Y?
What is the mgf of aX ?
Tuesday, 10 March 2009

22. MX+Y (t) = E(e )
(X+Y )t
= E(e e)
Xt Y t
= E(e )(e )
Xt Yt
= MX (t)MY (t)
MaX (t) = MX (at)
Tuesday, 10 March 2009

23. Y = a1 X1 + a2 X2 + · · · + an Xn
MY (t) = MXi (ai t)
If the X’s are iid, and the a’s are constant, then
MY (t) = MX (at) n
Tuesday, 10 March 2009

24. Your turn
Let X1, X2, …, Xn be iid random variables.
Xi
¯=
X
n
What is the mgf?
Tuesday, 10 March 2009

25. Convergence in P
Imagine you have a Bernoulli(p) process.
You can repeat the process as many
times as you like to generate X1, X2, …,
Xn. How could you use these X’s to figure
out what p is?
Tuesday, 10 March 2009

26. Convergence in P
It’s an intuitive answer (and correct), but
how can you check it?
Want to say something like:
n
1
lim Zn = p
Zn = Xi
n x=1 n→∞
But this doesn’t make sense. Why not?
Tuesday, 10 March 2009

27. lim P (|Zn − p| ≤ ) = 1
n→∞
∀ >0
Tuesday, 10 March 2009