The problem involves a random process defined as N(t, a) = At, where A follows a uniform continuous distribution over the interval (0, 3). The tasks include sketching sample functions for specific values of 'a' and finding the probability density function (PDF) of a transformed random variable.
The discussion is ongoing, with participants exploring different interpretations of the problem. There is no explicit consensus, but some productive insights have been shared regarding the nature of the random process and its implications.
Participants note the vagueness of the problem statement and question the clarity of the notation, which may affect their understanding and approach to the tasks.
cutesteph said:Homework Statement
Word for word of the problem:
Let N (t, a) = At be a random process and A is the uniform continuous distribution (0, 3).
(i) Sketch N(t, 1) and N(t, 2) as sample functions of t.
(ii) Find the PDF of N(2, a) = 2A.
Homework Equations
A pdf is 1/3 for x in [0,3]
The Attempt at a Solution
This question is very vague to me. I am not sure where to start.
haruspex said:I'm not familiar with this notation and terminology. I think it's saying that at time t you take a random sample x from A and get xt as the value of N. Is that right? I can only suppose that 'a' represents the random sample value, but that is strange notation since it is a function of t. Anyway, that leads to interpreting N(t, 1) as being the function of time you would get if the sample from A is always the value 1.
Correspondingly, N(2, a) is the r.v. obtained by taking samples from A and doubling them.
Does that all make sense?
Yes, I think that comes to the same as what I wrote, but better expressed.Ray Vickson said:My guess would be that this type of "random process" is hardly what we usually mean by that terminology: I guess we first choose a random realization 'a' of the random variable A, then for all time we have N(t,a) = a*t. That amounts to choosing a random slope for a line, but from then on having a deterministic line. Or so, that is what I read into the problem.
Ray Vickson said:My guess would be that this type of "random process" is hardly what we usually mean by that terminology: I guess we first choose a random realization 'a' of the random variable A, then for all time we have N(t,a) = a*t. That amounts to choosing a random slope for a line, but from then on having a deterministic line. Or so, that is what I read into the problem.