# If X(t) is gaussian process, How about X(2t)?

written as title,

1.
If X(t) is gaussian process, then

Can I say that X(2t) is gaussian process?

of course, 2*X(t) is gaussian process

2. If X(t) is poisson process, then

X(2t) is also poisson process?

Last edited:

mathman
In both cases you have a stochastic process where the element at a particular value of t has the specified distribution. Changing the scale factor doesn't affect that property.

Helolo
In both cases you have a stochastic process where the element at a particular value of t has the specified distribution. Changing the scale factor doesn't affect that property.

THANKS

Of course when ##X(t)## is gauss process, ##X(2t)## is also gauss cuz ##X(2t)=2X(t)## applied by ##f(\alpha x)=\alpha f(x)## .

Mark44
Mentor
Of course when ##X(t)## is gauss process, ##X(2t)## is also gauss cuz ##X(2t)=2X(t)## applied by ##f(\alpha x)=\alpha f(x)## .
I don't know anything about Gaussian processes, but I doubt very much that X(2t) = 2X(t). For most functions, ##f(\alpha x) \neq \alpha f(x)##.

What do you mean?

Mark44
Mentor
What do you mean?
What I mean is that, in general, functions aren't linear. Here are a few examples:
##\cos(2x) \neq 2\cos(x)##
##\sqrt{2x} \neq 2\sqrt{x}##
##\ln(2x) \neq 2\ln(x)##
##10^{2x} \neq 2\cdot 10^x##
etc.

berkeman
FactChecker
Gold Member
A Gaussian process, X(y), in variable y is just any process that is a normally distributed random variable for every value of y. It is not important the the variable y is time or not. If X(t) is a Gaussian process in time t, let y=2t, The process X(y) is a normally distributed random variable at y whose parameters, mean and variance are the same as X(t) where t=y/2 . So X(y), y=2t is also a Gaussian process. The Poisson process is similar, but not as simple. Since the Poison properties are very dependent on time t, changing to another variable y=2t is not so obvious. You should check if all the Poisson properties still apply with the new variable y.

Helolo
What I mean is that, in general, functions aren't linear. Here are a few examples:
##\cos(2x) \neq 2\cos(x)##
##\sqrt{2x} \neq 2\sqrt{x}##
##\ln(2x) \neq 2\ln(x)##
##10^{2x} \neq 2\cdot 10^x##
etc.
We are talking about Gauss process so the functions ##f## of it are linear

Mark44
Mentor
We are talking about Gauss process so the functions ##f## of it are linear

You just search on google, ok im done here. The end of reply

Mark44
Mentor
You just search on google, ok im done here. The end of reply
You made the claim - it's up to you to justify it.

PWiz
FactChecker