Distribution of the maximum of a RV

[yamdizzle]

I have a normally distributed rv,let be X_t, ~ (μ*t,t*σ^2)

what's the distribution of max(X_t) ?
how do we do this? I wanted to simulate but the more I simulate the more the values expand and explode.

Any help?

Or an easier question which can help me solve this. I have a joint cdf of (max(X),X) how can I get their joint pdf? I need to do the jacobian I think but not sure how.

 

[Stephen Tashi]

You have to explain what your notation " max(X_t)" means. It appears to be the maximum of a collection of things, but what things?

Perhaps you don't have "a normally distributed rv", but have a stochastic process instead. After all, if you only had one random variable in a sample, it would have one value, so that value would be the maximum.

 

[yamdizzle]

Yes this is stochastic. I will explain it more thoroughly:
It is a 2 step question I guess:

t \in[0,T]
X is a Brownian Motion (0, μ, σ^2)

M_T is the Max of X_t

I need to find the joint pdf of (X_T,M_T)

____
An easier question I guess
X is now has a drift 0. Therefore ~ (0, 0, σ^2)
find the joint pdf of (M_T - X_T, M_T)

I found the P(M_T > b | X_T =a) = exp(\frac{-2b*(b-a)}{T*σ^2} )
and P(M_T > b , X_T =a ) = \frac{1}{σ*sqrt(T)} * \Phi' ((\frac{a}{σ*sqrt(T)} )
where phi prime is the normal pdf
but not sure how to progress...
Any help would be appreciate. Sorry for not clarifying the question