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Homework Help: Applied Stochastic Processess?

  1. Oct 23, 2013 #1
    Random variable X is distributed according to Gaussian distribution
    P(x)= (1/sqrt(2πσ^2) * exp(- (x-μ)^2 / (2σ^2) )
    1. Calculate its first three moments
    2. What are the distributions of Y= X+b; Z=σX; W=X^2

    P(x)= (1/sqrt(2πσ^2) * exp(- (x-μ)^2 / (2σ^2) )
    the moment of first degree is:
    E(X) = Int (on |R of) x dP(x) = m

    1.2. moment of second degree:
    we have Var(X) = σ^2 = E(X^2) - E(X)^2 = E(X^2) - m^2
    so the moment of degree 2 is
    E(X^2) = σ^2 m^2

    1.3.moment of third order :
    as P is an even function :
    E(X^3) = Int ( on |R of) x^3 P(x) dx = 0
    and it 's the case for any odd moment n : n = 2k 1

    the gaussian characteristic is stable through linear transformation :
    Y = X b is gaussian
    ith the same σ
    and with an expectence of E(Y) = m b
    so Y is a N(m b ; σ)

    Z = σX is gaussian too :
    E(Z) = σm and Var(Z) = σ^2 Var(X) = σ^4
    Z is therefore a N(σm , σ^2) (σ^2 is its standard deviation

    W = X^2 follws a Khi-squared distribution law ;

    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 23, 2013 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    1.1. Determine the value of m, in terms of μ and σ.
    1.2. From σ^2 = E(X^2) - (E X)^2 we do NOT get E(X^2) = σ^2 * (E X)^2, which is what you wrote. Did you mean that, or was it a typo?
    1.3. P(x) is most definitely not an even function if μ ≠ 0. Start over.

    I don't know what you are doing in 2. You were asked about X+b but instead, you told us about Xb. Are you sure you were asked about σX? A more meaningful question would be about (1/σ)X. As for X^2: just apply the standard formulas you find in your textbook or course notes. It might be quite complicated; much easier would be the case of X^2 if we had μ = 0.
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