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Given the bivariate random variable (X,Y) with PDF: $ \begin{cases} \frac{2}{7}(2x+5y) & 0<x<1,\ 0<y<1 \\ 0 & otherwise \end{cases} $

  1. $ Let\ Z=\frac{Y}{X}. Compute\ the\ probability\ density\ function\ of\ Z. $
  2. $Compute \ P(X>\frac{1}{3}|Z=z) for \ z \in(0,\infty) $

I tried computing the Jacobian matrix but am not sure what to do then. Thanks

MilTom
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    Hint for 1. and 2.: Compute the joint PDF $f_{X,Z}$ of $(X,Z)$ using the Jacobian formula. Then the PDF $f_Z$ of $Z$ is given by $$f_Z(z)=\int_\mathbb Rf_{X,Z}(x,z)dx$$ and the answer to 2. is $$P(X>\tfrac13\mid Z=z)=\frac1{f_Z(z)}\int_{1/3}^\infty f_{X,Z}(x,z)dx$$ – Did Nov 08 '17 at 20:35

1 Answers1

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Using delta function approach ...

$$p(z')dz' = \delta(z'-z(x,y))p(x,y)dx dy$$

and hence

$$P(z''< z)=\int_0^z{p(z')}dz'$$

phdmba7of12
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  • and then, the RHS becomes $\int_0^zdz'\int_0^1dy\int_0^1dx\delta(z'-y/x)\frac{2}{7}(2x+5y)$ – phdmba7of12 Nov 01 '17 at 18:11
  • using the delta function of a function formula and integrating $dx$ we have $P(z"<z)=\int_{0}^{z}dz'\int_0^1dy\frac{2}{7}(2y/z'+5y)/|{-yln(y/z')}|$ – phdmba7of12 Nov 01 '17 at 18:30
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    do you know a simpler method since I have not studied the delta function appoach? – MilTom Nov 02 '17 at 09:59