Random process of uniform- graphing and pdf

[cutesteph]

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.

 

[Ray Vickson]

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.

You have both a and A; which one do you mean?

 

[haruspex]

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?

 

[Ray Vickson]

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?

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.

 

[haruspex]

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.

Yes, I think that comes to the same as what I wrote, but better expressed.

 

[cutesteph]

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.

That makes sense now.