on the probability density function

#1
Good morning to all. I have a question:
I have a data set X={x_1, x_2, ... , x_N} with probability density function P(x) (suppose continuous); if I operate on data with any mathematical function f(x) (square, power, exponential, trigonometric...), how is the new probability density function Q(x) related to the old P(x)?
Q(x) is the probability density function of data set Y={f(x_1), f(x_2), ... , f(x_N)}.

Thanks to all,
Statlinux
 

BGM

TS Contributor
#2
You should look for the Jacobian Transformation.

In the simplest case, assume that the inverse function of \( f \) exists,
and denote the inverse be \( g \), i.e. \( Y = f(X) \iff X = g(Y) \)

And assume that \( f \) is monotone and differentiable.

Then
\( Q(y) = P(g(y))\left|\frac {\partial g(y)} {\partial y}\right| \)
 
#4
You should look for the Jacobian Transformation.

In the simplest case, assume that the inverse function of \( f \) exists,
and denote the inverse be \( g \), i.e. \( Y = f(X) \iff X = g(Y) \)

And assume that \( f \) is monotone and differentiable.

Then
\( Q(y) = P(g(y))\left|\frac {\partial g(y)} {\partial y}\right| \)
I looked for it, and I found almost all I need. I have one more question:
if I have random function x(t), with probability density P(X), and I integrate or differentiate x(t) with respect to time variable, how does P(x) transform? Is it possible to answer?

Thanks,
StatLinux
 

BGM

TS Contributor
#5
Seems like you are talking about the stochastic processes.

Actually doing integration is just somehow "summing up" the processes.

In some nice case, e.g. \( W(t) \sim \mathcal{N}(0, t)\) is a Wiener process,

then \( \int_0^T \sigma(t) dW(t) \sim \mathcal{N}\left(0,
\int_0^T \sigma(t)^2dt \right) \)

but in general the distribution could be very complex.

And usually you need much stronger conditions such that the process
is smooth enough to be differentiable.
 
#6
Yes you are right. I'm talking about stochastic processes. Unluckly I'm not familiar to this branch. Anyway I'm happy to know that there is possibility to compute probability density function of composed functions.

My aim is not only that to operate on single numbers, but o functions:
integrare, differentiate, sum, product, fraction of functions.

Thanks a lot,
StatLinux