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Distribution of the maximum of a RV

  1. Dec 1, 2011 #1
    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.
  2. jcsd
  3. Dec 1, 2011 #2

    Stephen Tashi

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    Science Advisor

    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.
  4. Dec 1, 2011 #3
    Yes this is stochastic. I will explain it more thoroughly:
    It is a 2 step question I guess:

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