----------------------------------------------------------------------
Professor:
could you explain what is the difference between Ak and Ak' as they are
used in the proof of property(1.4) on page 6 in the textbook?
thanks.
The Ak's are arbitrary events - the Ak primes are mutually incompatible.
They do not contain outcomes of any Al, l <= k-1.
----------------------------------------------------------------------------
I enjoyed your brief philosophic introduction in
Lesson 1 and a question I wanted to ask you was
whether your study and experience leads you to believe
that either all processes are deterministic, and that
probability is just a model to reduce uncertainty, or
there alternative view that there are some irreducibly
random processes.
Thanks.
My view is that when the complexity of phenomena being investigated
is very large, stochastic models are often effective in predicting
behavior.
I am not taking a position about what things really are --- deterministic
or stochastic. We only deal with models. Thus, I do argue that there is
nothing intrinsically random.
What I would like you to appreciate, however, is that by using
stochastic models we end up with a calculus of probabilities.
This calculus is purely "deterministic". In the introduction on page x
there is a sentence " ... the modeling power of Markov Chains may well
be compared to that of ordinary differential equations". We will be
giving a clear meaning to this very important statement towards the
end of the semester.
----------------------------------------------------------------------------
If I have questions about the
course material. Should I e-mail them to you or do you
have a TA you'd rather have work with students?
Please feel free to e-mail me questions if they pertain to the material
covered in the class. Questions regarding the homework, problem sets,
etc., should be addressed directly to the TA, Raul Rodriguez-Esteban.
----------------------------------------------------------------------------
(There is a difficulty expressing equations in email, so if prefer
me to fax the handwritten questions, please let me know.)
e-mail is fine
-------------------------------------------------------------------
p.3
The first equation with summation. Is "k=0" a mistake for "k=1"?
yes, k=1
--------------------------------------------------------------------
p.9
Example 2.2 The 2nd line in the description. I'll use () instead
of subscript:
{X(1)=a(1), ..., X(k-1)=a(k-1), X(k)=a(k)} for all possible values
of (a(1), ..., a(k-1)) <-- Why is this last element a(k-1)? Is
this a mistake for a(k)?
This is correct. In order to get P(X(k)=a(k)) you have to sum
P(X(1)=a(1), ..., X(k-1)=a(k-1), X(k)=a(k)) over all a(1), ... , a(k-1).
------------------------------------------------------------------------
p.17
Example 3.2 I assume that P(S(n)=k) is the probability that the
number of heads = k. If so, the number of tails is n-k. I'll use
^ to express the power. Why is this probability
n 1 n
( ) --- i.e. ( ) p^n
k 2^n k
instead of
n
( ) p^n (1-p)^(n-k)
k
?
The expressions are the same because p=1/2.
-------------------------------------------------------------------------
p.18
From line #2 to #3, Will you please show me why the summation
becomes 2^(n-1) ? (Or referring any book or website that has
explanation would be fine too.)
The expression in the parenthesis is just (1+1)^(n-1)
--------------------------------------------------------------------------
p.15 A line above section 3.2
I find many probability books simply clarify discrete or continuous
random variables. This book use the term "absolutely continuous".
Does this imply there are such kinds as "not" absolutely continous?
The answer is yes. A function is absolutely continuous IFF it can be
represented as an indefinite integral (the definition of an absolutely
continuous function can be found in a book in Real Analysis (Royden,
for example).
----------------------------------------------------------------------
p.20 Eq.(3.22)
The power to the last term, shouldn't k-summation(aj) be n-summation
(aj) ?
Why n? The probability is determined for the first k events. There is
no n in equation (3.22).
--------------------------------------------------------------------
p.20 Eq. in the bottom line
T=inf{ n>=1; Xn = 1 } --- Considering from the 1st line in p.21, I
think this is a mistake for Xn = 0, is it correct?
No, X(n) = 0 - this is the case in which no ones will occur.
---------------------------------------------------------------------
p.21 Poisson
I don't understand why pN(0) goes to exp(-a) as N goes to
infinity. Will you please explain?
(1+1/n)^n goes to e in the limit. This is a classical result in analysis.
----------------------------------------------------------------------
p.21 Poisson, after eq.(3.27), Poisson law of rare events
I don't quite understand intuitively why obtaining tails is an
increasingly rare event as N goes to infinity. If we keep flipping
a coin a large number of times, I would imagine heads and tails
would appear equally likely. Will you please help what I'm
misunderstanding?
The probability of tails P(X(n)=1) goes to zero as N goes to infinity
(by construction). It is a rare event in the sense that the probability
distribution is vanishingly small for large N. However, S(N) tends to a
Poisson random variable. Note again the abuse of language - there is
nothing Poisson about a random variable - the distribution is Poisson.
-----------------------------------------------------------------------
The sum of several exponential rvs is not another exponential, but is a
gamma rv. Why then does problem 1.4.3 ask us to show that the output
vector {S_n} is a sequence of exponential rvs? Since S_n is the sum of
several exponentials, S_n isn't an exponential rv (unless the summation is
over only one rv).
The result you quote applies to a fixed (as opposed to random) sum of
random variables.
-----------------------------------------------------------------------
About assigned hw question that specified a circle x^2+y^2 = 1 and
R.V.'s X(w)=x and Y(w)=x, and prove that X, Y are not independent
(sorry i don't have the book with me, it's 1.2 or 1.3). Is Y(w)=x a
typo? It would make more sense to have Y(w)=y?
yes, it is a typo.
-------------------------------------------------------------------
Also, I'm a first year student attending Columbia and I'm not
familiar with the CVN/long distance learning program. It appears
that people do not ask questions in class, but in emails. I just
wonder if it's ok to ask in class, since I saw microphones hanging
from the ceiling for the students... just wondering.
Please feel free to ask question in class
--------------------------------------------------------------------
Another thing is that will we get in-class exams? I'm asking because
CVN students can only take an exam using emails?
We will have in-class exams. CVN student exams are handled separately.
-----------------------------------------------------------------------
Is (4.15) holds for any two R.V. or they have to be independent? Is the
"X" should be "Y" in the paragraph right below (4.15)?
Yes, Y and Z are assumed to be independent in the derivation of (4.15).
And yes, read Y instead of X below (4.15).
-----------------------------------------------------------------------
About a question you responded to a while ago, quoted below.. I'm
wondering if you could please elaborate on the equations used to reduce
the summation to (1+1)^(n-1)?
A book does not come to mind but here is a short proof by induction
for (1+1)^n.
First, equality is true for n=1. Assume that equality is true for
n=k. Let's show that it is also true for n=k+1. Note that
(1+1)^(k+1) = (1+1) times (1+1)^k. Arrange the known terms of
(1+1)^k in two rows but shift the terms of the second row by one
to the right. Now add the terms in the same column.
Note that (n over j) + (n over j-1) = (n+1 over j).
------------------------------------------------------------------------
p.18
From line #2 to #3, Will you please show me why the summation
becomes 2^(n-1) ? (Or referring any book or website that has
explanation would be fine too.)
The expression in the parenthesis is just (1+1)^(n-1)
------------------------------------------------------------------------
At the bottom of pg 54, (1.3) is stated and then proven on pg 55 and 56.
Disregarding the proof, can't you just say that p_i,i+1 is just equal to
P(U1 > i+1 | U1 > i) based on the diagram, and that this is equal to
P(U1 > i+1)/P(U1 > i)?
While the statement is correct how do you know that the assertion is true?
For example, assume that the U(n)'s are not independent. Would the same
result hold? Not in general of course although the diagram appears to be
the same.
---------------------------------------------------------------------------
p.33 Eq.(5.9) characterstic function:
What is T and where did it come from? (whereas in your lecture #3
it was defined as psi_X(u) = E[exp(iuX)] and not u^T)
T stands for "transpose" - u is defined in general as a column vector.
u^T is a row vector.
In class I assumed for simplicity that u is a scalar (I believe). The
formalism is the same.
---------------------------------------------------------------------
p.33 Example 5.4 (i) Gaussian
Isn't sigma missing on the right-hand side of the pdf? i.e. 1/(sqrt
(2*pi) * sigma) ?
Yes, you are right. Good eye!
---------------------------------------------------------------------
| Overview | Outline | Midterm | Final | FAQ |