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.
A random process of uniform graphing is a mathematical model that represents a system with equal probability of producing any outcome within a given range. It is often used to study and analyze phenomena that involve a large number of random variables.
A random process of uniform graphing is typically represented using a probability density function (PDF) which shows the probability of each possible outcome occurring. This function is often depicted as a graph or curve.
A random process of uniform graphing assumes that all outcomes within a given range are equally likely, while a normal distribution assumes that outcomes follow a bell-shaped curve with most values clustered around the mean. Additionally, a normal distribution has a finite mean and variance, while a random process of uniform graphing has infinite mean and variance.
A random process of uniform graphing is commonly used in fields such as statistics, physics, and engineering to model and study systems with random behavior. It can also be used in simulations, computer algorithms, and risk analysis.
One limitation of a random process of uniform graphing is that it assumes all outcomes are equally likely, which may not be true in real-world scenarios. It also does not take into account correlations between variables and cannot accurately model complex systems with non-linear relationships.