Homework 3 is now posted to the course website. It is due next Friday at 1:30pm. Good luck!

Of the four questions on the HW, the 1,2, and 4 you should be able to attack now. The third one is probably best done after Monday's lecture. (4 is related to what we'll do in class on Wed, but doesn't require anything that I'll teach on Wed, so you should be able to do it now.)

As always, if you have any questions about the HW, or just want to point out typos, please post them as comments on this post and then the TAs/prof will respond ASAP.

## Friday, January 20, 2012

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Please note that a typo in Problem 4(c) has been fixed. The updated pdf will be synced on the course website soon. The change is as follows: In the algorithm provided in 4(c), an intermediate node on the ring that receives a packet from the middle node B will

ReplyDeletealwaysforward it along the ring.Hey, I'm having trouble with 4a), calculating P(l, k). This seems really difficult. I can get values for P(1,k) and P(l,k) where l > k. However, even for another simple value like P(2, k) for some given k, I can't calculate this. In this case there are two options, but the probabilities aren't independent so I can't get an expression. Do you have any tips on this problem? It's tough because the rest of 4) builds on this part. Thanks,

ReplyDeleteKudos on starting early! I don't want to spoil it too much for others who'd like to think more about this on their own, and I didn't get what you meant by two options, so I'll just answer your question with another question (a small hint) - for l<k, how many times do you think the shortest path will visit the central node B? Please don't answer this here - to continue this discussion, consider emailing me, or waiting till Wednesday evening!

DeleteHey guys, hope you're having a good day!

ReplyDeleteI am a little confused about the plots (problem 3) ...

Using the Mathematica 8 reference,

The Weibull Distribution is defined as such:

WeibullDistribution[alpha, beta]

represents a Weibull distribution with shape parameter alpha and scale parameter beta.

The Pareto Distribution is as such:

ParetoDistribution[k, alpha]

represents a Pareto distribution with minimum value parameter k and shape parameter alpha.

In this problem we are asked to make plots with a Weibull distribution of alpha = .3. What should we use as the beta value, 1? (This is the expected value, although I am not sure about how this will be the scaling factor)

Also, we are supposed to make plots of a Pareto distribution with alpha 1.5, and another one with alpha .5 and scale parameter 1/3.

I though only the Weibull distribution had a scale parameter ... also, what should be the minimum value k for the Pareto distribution?

Thanks!

Actually, I understand we have to SET the Expected Value to be 1 so we pick our parameters accordingly!

DeleteHowever, I am confused as to what is the scale parameter of a Pareto Distribution.

What is the scale parameter for a Pareto Distribution?

ReplyDelete'k' in your notation above is the scale parameter. See the Wikipedia article on Pareto distribution (the first entry in the table summary on the right side lists the parameters and what they are called).

DeleteI'm a little confused about question 2(c) where it says "contrast this with the result of part (a)." The answer to part (a) does not define a probability distribution. You can however let X be the probability of typing any c-letter word and you get Pr[X = x] = n^c * [answer to (a)]. We can then say something about how heavy the tail of X's distribution is. Is this what you want us to do? Thanks!

ReplyDeleteThe answers to both part (a) and (c) define two probability distributions on the entire dictionary of possible words (i.e., on \Sigma^*, where \Sigma is the alphabet containing n letters). For example, if your alphabet is {0,1}, then Pr[typed word = 000110] = answer to (a) with c=6.

DeleteWhat you are suggesting is also acceptable. In fact, I would say your idea is more relevant and actually leads to a meaningful contrast.

DeleteFor those using Mathematica, you can use the following arguments for labeling your plots:

ReplyDeleteAxesLabel, PlotLegend, LegendPosition (just search Mathematica Help for these keywords to find out the syntax).

PlotLabel is also useful to add a title to your graphs

Delete