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Mean and autocorelation

  1. May 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Let V₁,V₂,r be independant random variables where V₁,V₂ are gausian with the same distribution and r is uniformly distributed in [0,1] and

    X(t)=V₁I(r<t)+V₂I(r≥t)

    a)find the mean of this process:
    b) find the autocorelation function of this process

    2. Relevant equations



    3. The attempt at a solution
    a) EX(t)=E[V₁I(r<t)+V₂I(r≥t)] by independance of V₁,r and V₂,r
    =EV₁∗E I(r<t)+EV₁EI(r≥t)=
    μ(1-t)+μ(t)=μ

    b) i am stuck on this bit:
    R(t,s)=E[V₁I(r<t)+V₂I(r≥t)∗V₁I(r<s)+V₂I(r≥s)]
    =E[V₁²I(r<t)I(r<s)]+E[V₂²I(r≥t)I(r≥s)]+E[V₁I(r<t)V₂I(r≥s)]+E[V₂I(r≥t)V₁I(r<s)]
    =σ²P(r>t,r<s)+σ²P(r≥t,(r≥s)+μ²P(r<t,r≥s)+μ²P(r≥t,r<s)
     
  2. jcsd
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