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

[admin]

Thank you Dr. Krzysztof Ostaszewski for suggesting we post some of your sample problems. . Watch for further exam problems as he makes them available to us.

[admin]

March 12th Problem

http://www.math.ilstu.edu/krzysio/3-...O-Exercise.pdf

[admin]

March 19th Problem

http://www.math.ilstu.edu/krzysio/3-...O-Exercise.pdf

[admin]

March 26th Problem

http://www.math.ilstu.edu/krzysio/3-...O-Exercise.pdf

[admin]

April 2nd Problem

http://www.math.ilstu.edu/krzysio/4-2-5-KO-Exercise.pdf

[krzysio]

April 9 exercise

Here is another practice problem:
http://www.math.ilstu.edu/krzysio/4-9-5-KO-Exercise.pdf
Yours,
Krzys'

[krzysio]

April 16 exercise

http://www.math.ilstu.edu/krzysio/4-16-5-KO-Exercise.pdf

Enjoy,
Krzys'

[krzysio]

April 23 exercise

Some discrete probability basics:

http://www.math.ilstu.edu/krzysio/4-23-5-KO-Exercise.pdf

Enjoy.
Yours,
Krzys'

[engelbmj]

question about the 3-19-5 problem

I have a question about the 3-19-5 problem posted. In the 4th line of the solution:

Cov(X1-aX2-bX3,X2) = Cov(X2,X1) - aVar(X2) - bCov(X3,X2)

I'm not sure how you got from left hand side to the right hand side of the equation. I've never seen an covariance example like this. I'm familiar with the equation for the covariance of 2 variables, but not of the variance of an expression and a variable. Could you generalize the rule for me? Thanks very much.

[wat]

I have a question about the 3-19-5 problem posted. In the 4th line of the solution:

Cov(X1-aX2-bX3,X2) = Cov(X2,X1) - aVar(X2) - bCov(X3,X2)

I'm not sure how you got from left hand side to the right hand side of the equation. I've never seen an covariance example like this. I'm familiar with the equation for the covariance of 2 variables, but not of the variance of an expression and a variable. Could you generalize the rule for me? Thanks very much.

I think you can think of covariance like an inner product, or, if you like, the FOIL method. So, in this case,
Cov(x1 - ax2 - bx3, x2)
= Cov(x1,x2) - Cov(ax2,x2) - Cov(bx3,x2)
= Cov(x1, x2) - a*Cov(x2,x2) - b*Cov(x3, x2)
= Cov(x1, x2) - a*Var(x2) - b*Cov(x3, x2)

[krzysio]

Covariance

Wat is exactly right. Or look at it this way: Covariance is a linear operator in each of its variables (This problem's purpose was to teach you that, comes in handy in all covariance problems on the test)

Cov (X + Y, W) = Cov (X, W) + Cov (Y,W)
Cov (X, Y+W) = Cov (X, Y) + Cov (X,Y)
Cov (aX, W) =a Cov (X, W)
Cov (X, aW) =a Cov (X, W)

Does this help?

Yours,
Krzys'

[krzysio]

April 30 exercise

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

Enjoy!

Yours,
Krzys'

[krzysio]

Weibull distribution

Yes, Weibull is on the test. And it is kind of hard to memorize all the distributions. So, to help you memorize the crucial item about the Weibull distribution, consider this exercise:

The time to failure X of an MP3 player follows a Weibull distribution. It is known that and that the probability that this player stil works after 3 years is 1/e, while the same probability for 5 years is 1/(exp(4)). Find the probability that this MP3 player is still functional after 4 years.

exp(4) means just e to the power 4.

In other words, what is the survival function of the Weibull distribution, and can you memorize it?

Yours,
Krzys'

[krzysio]

Exercise for April 30, 2005

I have now posted this exercise with a solution:

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

Enjoy.

Yours,
Krzys'

[krzysio]

Exercise for May 7, 2005

I have now posted this exercise with a solution:

http://www.math.ilstu.edu/krzysio/5-7-5-KO-Exercise.pdf

Enjoy.

Yours,
Krzys'