Weekly Practice Exam "P" Problems courtesy of Dr Krzysztof Ostaszewski

[krzysio]

May 14 exercise for exam P

I have posted a new exercise for exam P:

http://www.math.ilstu.edu/krzysio/5-14-5-KO-Exercise.pdf

This is in response to questions raised by some concerning transformations of random vectors, and sums of random variables, convolution and derivation of the PDF of the result of those operations. The exercise is given with three different solutions, showing three different approaches you can use for sums or differences of random variables: transformation of random vectors, convolution, or the CDF technique. It may feel a bit hairy, but it is meant to show you all of these techniques and help you deal with this kind of problems in every possible way. I hope you will enjoy.

Yours,
Krzys' Ostaszewski

[jessie]

April 30 problem

We are asked to find the excess of Pr(X>11/4) over its bound.
I don't understand why in the solution we apply the chebyshev's inequality to Pr((X-3/4|>30/16). And then find the prob of Pr(X>21/8), why do we need to calculate this probability?

Shouldn't we rewrite the original Pr(X>11/4) to be
Pr(|X-3/4|>8/4)=Pr(X>11/4)+Pr(X<-5/4)=Pr(X>11/4)

Thanks.

[krzysio]

Chebyshev's Inequality

Dear Jesse:

You are absolutely right, I messed up. I posted a corrected version now:

http://www.math.ilstu.edu/krzysio/4-30-5-KO-Exercise.pdf

My apologies for this mess.

Yours,
Krzys'

[davenpoa]

Variance/Covariance Matrix

I have never seen a variance/covariance matrix before. Can you explain what the numbers in the matrix mean for the March 19, 2005 problem?

[wat]

I have never seen a variance/covariance matrix before. Can you explain what the numbers in the matrix mean for the March 19, 2005 problem?

In a covariance matrix, the (i,j)th entry of the matrix = Cov(Xi,Xj).

So, by properties of covariance and variance:

1.) The matrix is symmetric (since Cov(Xi,Xj) = Cov(Xj,Xi)).
2.) Cov(Xi,Xi) = Var(Xi).

[Karen]

variance/covariance matrix

Would you mind explaining the variance/covariance matrix that I have seen come up in some of your problems? My study guide did not include that.
Thanks!

[Karen]

April 30th problem

I have now posted this exercise with a solution:

http://www.math.ilstu.edu/krzysio/4-30-5-KO-Exercise.pdf

Enjoy.

Yours,
Krzys'

I was able to do this problem all the way until I needed to find the exact probability. I am not sure how the exact probability was found (specificially the jump from Pr(x>2.686) to 3/4e^-2.686. ) This may just be an equation that I am unaware of.

Thanks for your help!

[wat]

Would you mind explaining the variance/covariance matrix that I have seen come up in some of your problems? My study guide did not include that.
Thanks!

In a covariance matrix, the (i,j)th entry of the matrix = Cov(Xi,Xj).

So, by properties of covariance and variance:

1.) The matrix is symmetric (since Cov(Xi,Xj) = Cov(Xj,Xi)).
2.) Cov(Xi,Xi) = Var(Xi).

Was there any part that wasn't clear?

[krzysio]

Explanations

1. Variance-covariance matrix -- Wat's explanation is perfect. Did it help? It is troubling, though, that your study material did not include this -- this is a part of the study of multivariate distributions, and you do need to study this for the test. You might want to reconsider your study materials.

2.The distribution in the problem is basically exponential with weight 3/4 (and a point-mass with weight 1/4 at 0). The point mass does not matter for this probability, as we are estimating survival function for a positive value. The PDF, CDF and survival function of a mixed distribution is the weighted average of individual pieces' PDF, CDF, survival function (respectively). So the survival function is
1/4 * point mass survival function (which is zero at this point) +
3/4 * exponential survival function.

By the way, in the minds of examiners, an actuary can state the survival function for exponential distribution (exp (-x/mean)) i n s t a n t l y. You need to make sure that this is how well you know it.

Yours,
Krzys'

[mamadou]

Help on conditioanal mixture

Dr krzysztof I really need your help on conditional mixture,mixed with total probability or Baye's law.I just don't understand the principle.

[davenpoa]

Pareto

I was wondering if someone could explain when you use Two Parameter Pareto vs One Parameter Pareto if a problem only says it's Pareto.

[krzysio]

Pareto

The syllabus for exam P only says that the basic Pareto distribution is on the exam. Thus you do not need to study its variations.
Yours,
Krzys'

[krzysio]

Mixed distributions

Dr krzysztof I really need your help on conditional mixture,mixed with total probability or Baye's law.I just don't understand the principle.

Mixed distributions are commonly on the first and third exams so you really should get a book that covers this and study it. The manual I wrote covers it. The idea is that you take a weighted average of two (or more) probability distributions (NOT random variables, but their probability distributions) with positive weights that add up to one (those weights are basically probabilities of ending up in one of those distributions). From this it follows that the mixed distribution's PDF, CDF, survival function and all moments are analogous weighted averages of the pieces that it consists of.

Yours,
Krzys' Ostaszewski

[krzysio]

More on mixed distributions

Dr krzysztof I really need your help on conditional mixture,mixed with total probability or Baye's law.I just don't understand the principle.

As for "conditional mixture mixed with Bayes' Law," you really have to explain to me what you mean by the question. Please help me understand your question.

Yours,
Krzys'

[davenpoa]

The syllabus for exam P only says that the basic Pareto distribution is on the exam. Thus you do not need to study its variations.
Yours,
Krzys'

In some books that is the one paramenter and in other cases it seems to be the two parameter. I am guessing that you are refering to the two parameter distribution? Also do you think this exam will have more stuff like the 110 exam? Bivariate Normal and lognormal were on the list of things to know but I don't really know what kind of questions they would ask, do you have any suggestions?

Thanks,
Andrea