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Math Help - Simple problem

  1. #1
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    Simple problem

    I'm stuck but here's what I have so far. I'm trying to find the probability that S is bigger than T.

    T \sim N(18,4), S \sim N(20,1)

    Pr \{S>T \} = Pr \{ \frac{S-20}{1} > \frac{T-20}{1} \}

    =Pr\{N(0,1)>T-20\}

    =Pr\{T-20<N(0,1)\}

    \mbox{Let }X=T-20 \mbox{ then } X \sim N(18-20,4) \sim N(-2,4)

    \Rightarrow Pr\{T-20<N(0,1)\}

    =Pr\{X<N(0,1)\}

    =Pr\{N(-2,4)<N(0,1)\}

    Where do I go from here?
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  2. #2
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    Quote Originally Posted by financial View Post
    I'm stuck but here's what I have so far. I'm trying to find the probability that S is bigger than T.

    T \sim N(18,4), S \sim N(20,1)

    Pr \{S>T \} = Pr \{ \frac{S-20}{1} > \frac{T-20}{1} \}

    =Pr\{N(0,1)>T-20\}

    =Pr\{T-20<N(0,1)\}

    \mbox{Let }X=T-20 \mbox{ then } X \sim N(18-20,4) \sim N(-2,4)

    \Rightarrow Pr\{T-20<N(0,1)\}

    =Pr\{X<N(0,1)\}

    =Pr\{N(-2,4)<N(0,1)\}

    Where do I go from here?
    Are S and T independent? I suppose so. Then you should consider the random variable W = S - T and calculate Pr(W > 0).

    Quote Originally Posted by Wikipedia at en.wikipedia.org/wiki/Normal_distribution
    if X1, X2 are two independent normal random variables, with means μ1, μ2 and standard deviations σ1, σ2, then their linear combination will also be normally distributed:
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  3. #3
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    Wow, why couldn't I think of that myself?

    So now I have:

    \mbox{Let }X=S-T

    X \sim N(20-18, 1+4) \sim N(2, 5)

    Pr\{S>T\} = Pr\{W>0\}= Pr\{ \frac{W-2}{\sqrt{5}}>\frac{0-2}{\sqrt{5}} \}

    =Pr\{N(0,1) > \frac{-2}{\sqrt{5}} \}=\Phi(\frac{-2}{\sqrt{5}})

    Is this correct?
    Last edited by financial; October 17th 2010 at 04:46 PM. Reason: Corrected typo.
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  4. #4
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    On a side note, what if they're correlated?

    Let's say the covariance between S and T is 1.2. Now, how do I calculate the probability that S is greater than T?
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  5. #5
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    Quote Originally Posted by financial View Post
    On a side note, what if they're correlated?

    Let's say the covariance between S and T is 1.2. Now, how do I calculate the probability that S is greater than T?
    Then obviously you need to review the difference of two correlated normal distributions: http://www.stanford.edu/~srabbani/bivariate.pdf
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  6. #6
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    Okay this time I get

    [LaTeX ERROR: Convert failed]

    And continue the same calculation as previously.
    Thank you for your help.
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