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Multivariate normal distribution , conditional probability problem

  1. Jun 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Given X,Y RV both have normal distribution with:
    [itex]μ_{x}=6[/itex],[itex]μ_{y}=4[/itex],[itex]σ_{x}=1[/itex],[itex]σ_{y}=5[/itex],ρ=0.1

    a. are X,Y independent?
    b. find P(X≤5)
    c. find P(Y≤5|X=5)

    2. The attempt at a solution


    a. no -> ρ=0.1 -> cov(X,Y)≠0

    b. define Z=[itex]\frac{X-6}{1}[/itex] ; Z~N(0,1)
    so P(X≤5) = P(Z≤-1) = [itex]\Phi(-1)[/itex]

    c. Here i have my difficulty were i able to calculate and i know :
    μ=[6 4] , Ʃ=[1 0.5,0.5 25] cov matrix , [itex]Ʃ^{-1}[/itex]= [[itex]\frac{100}{99} \frac{-2}{99}, \frac{-2}{99} \frac{4}{99}[/itex]] inverse matrix.


    just don't how to deal with the conditional question...
     
  2. jcsd
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