pdf question

Discussion in 'Probability' started by B, Jul 2, 2010.

  1. B

    B Guest

    Have:
    X = requests per second with E[X]
    Y = resources used per request with E[Y]
    They are independent.
    f(x) is probability density function of E[X]
    g(y) is probability density function of E[Y]

    Question:
    How can h(f(x),g(y)) pdf of E[X*Y] be derived from f(x) and g(y)? For
    example, for E[X+Y] h(x,y) is a convolution of f(x) and g(y).

    In the end what I am really looking for is: given the pdf for arrivals
    and pdf of resources per arrival, how can I easily/quickly get the
    additional combined pdf/cdf s.t. I can say with 95% confidence that
    the total resources consumed by all arrivals is less than x. FWIW, I
    am working in the discrete world.
     
    B, Jul 2, 2010
    #1
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  2. f is pdf of X, not of E[X]
    here the same
    again, h is the pdf of X*Y, not E[X*Y]. And h depends on one parameter,
    not on two.
    Here it is similar. For the sum you have

    h(t) = pdf[X+Y](t) = int[0..t] f(t-s) g(s) ds

    and for the product

    h(t) = pdf[X*Y](t) = int[0..oo] f(t/s) g(s) ds

    For the expectation you have E[X*Y] = E[X]*E[Y].
    For the cdf you have

    cdf[X*Y](t) = int[0..oo] cdf[X](t/s) pdf[Y](s) ds

    Of course by the symmetry, you can also exchange the roles of X and Y if
    this makes it easyer to calculate.

    In the discrete case you have

    P(X*Y <= n) = sum[k=1..oo] P(X <= n/k) P(Y = k)

    Here, you get a finite sum, if P(X = 0) = 0 holds. If not, but
    P(Y=0) = 0 holds, exchange the roles of X and Y. And

    P(X*Y = n) = sum[k|n] P(X = n/k) P(Y = k) .

    Cheers
    Bastian
     
    Bastian Erdnuess, Jul 2, 2010
    #2
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  3. B

    Henry Guest

    You can actually deal with the zero case as

    P(X*Y=0) = P(X=0) + P(Y=0) - P(X=0,Y=0)
    = P(X=0) + P(Y=0) - P(X=0) P(Y=0)

    and

    P(X*Y <= n) = P(Y=0) + sum[k=1..oo] P(X <= n/k) P(Y = k)
     
    Henry, Jul 2, 2010
    #3
  4. B

    danheyman Guest

    What you DON'T want is the distribution of X*Y. Let Y_i be the
    resources needed by the ith arrival. The total number of resources
    needed is S=Y_1 +Y_2 +...+Y_X, where the Y's have density g. Assuming
    the Y's are iid and independent of X, the probability generating
    function (since you say the RVs are discrete) of the distribution of S
    is easily obtained in terms of PGFs of f and g (look up compound
    distribution). Numerical inversion may be needed, but in some special
    cases, the PGF can be inverted analytically (X is Poisson and Ys are
    Poisson or geometric if memory serves).
     
    danheyman, Jul 7, 2010
    #4
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