How can I determine the random variables for this problem?

[Boltzman Oscillation]

Homework Statement

Let N1 be the number of web page requests arriving at server in 100-ms period and let N2 be the number of Web page requests arriving at a server in the next 100-ms period. Assume that in a 1-ms interval either zero or one-page request takes place with respective probabilities 1-p = 0.95 and p =0.05, and that the requests in different 1-ms intervals are independent of each other.

a) Describe the underlying space S of this random experiment and show the mapping from S to Sxy, the range of the pair (X, Y).

Relevant Equations

none

So i first need to come up with the sample space, X, and Y.
Well I would guess that the random variables here are N1 and N2 and thus X = N1 and Y = N2. Now i need to make these random variables a function of L. I don't know what L should be but I would guess it is the outcome of a 1ms interval? I am completely lost. Could someone steer me towards the correct direction?

 

[FactChecker]

Where did L come from? I don't see it mentioned anywhere in your description of the problem. What is it? Is it a length of time, like 100ms?

 

[Boltzman Oscillation]

Where did L come from? I don't see it mentioned anywhere in your description of the problem. What is it? Is it a length of time, like 100ms?

Oops sorry.

A random variable X is a function that assigns a real number, to each outcome L in the sample space of a random experiment.

I guess L would be the outcome of 1ms interval n this case? Right?

 

[Ray Vickson]

So i first need to come up with the sample space, X, and Y.
Well I would guess that the random variables here are N1 and N2 and thus X = N1 and Y = N2. Now i need to make these random variables a function of L. I don't know what L should be but I would guess it is the outcome of a 1ms interval? I am completely lost. Could someone steer me towards the correct direction?

Start with much smaller problems, and study them until you understand them.

If you have one single 1ms interval, and we record the outcomes as 0 = "no request in that interval" and 1 = "one request in that interval", the sample space S would consist of two possible outcomes: S = {0,1}. What are the probabilities on this sample space? That is, what are P(0) and P(1)? The problem data tell you that.

Now go to two 1ms intervals, and use labels 0 and 1. Now the sample space would be S={(0,0), (0,1), (1,0), (1,1)}. Each of the four resulting pairs (i,j) tell us about request/non-request in period 1 and then request/non-request in period 2. Given the assumptions given in your problem statement, you ought to be able to write down the four probabilities P(0,0), P(0,1), P(1,0), P(1,1). In addition, you ought to be able to figure out such probabilities as P(no requests), P(exactly one request) and P(two requests). In the notation of your problem these would be P(N=0), P(N=1) and P(N=2).

Now all you need to is extend this to the case of 100 1ms intervals. In this case the sample space is so large that you would spend several years writing it all down, but you can describe it without too much trouble. You can also figure out what would be the probability of each separate outcome, so you could figure out P(o_1,o_2, ... , o_100), where o_i = 0 or 1 is the outcome in the ith 1ms interval; this probability will have a simple formula that depends only on n = number of 1s and m = 100-n = number of 0s.

After you have done that you should then be able to figure out quantities like P(N=k), because all the outcomes having k requests and 100-k non-requests have the same probability. That means that you need to figure out how many of the outcomes have counts k and 100-k.

I would not necessarily expect you to easily and quickly figure out all this for yourself, but it is in every probability textbook and in thousands of web pages. For example, try googling "coin-tossing probabilities". While you are not dealing with coin tosses here (which have p = 1/2) the logic is similar. If you look for material on "biased coin tossing" you ought to find much of what you need.