Mathematics, 10.03.2020 00:20 duplerk30919
4.52 Random variables X and Y follow a joint distribution f(x, y) = 2, 0 < x ≤ y < 1, 0, otherwise. Determine the correlation coefficient between X and Y .
Answers: 1
. By the law of total probability,
![E[X]=\displaystyle\int_{-\infty}^\infty x\,f_X(x)\,\mathrm dx=\int_0^12x(1-x)\,\mathrm dx=\frac13](/tpl/images/0539/2487/98c0c.png)
![E[Y]=\displaystyle\int_{-\infty}^\infty y\,f_Y(y)\,\mathrm dy=\int_0^12y^2\,\mathrm dy=\frac23](/tpl/images/0539/2487/59552.png)
![V[X]=E[(X-E[X])^2]=E[X^2]-E[X]^2](/tpl/images/0539/2487/9ca41.png)
![E[X^2]=\displaystyle\int_{-\infty}^\infty x^2\,f_X(x)\,\mathrm dx=\int_0^12x^2(1-x)\,\mathrm dx=\frac16](/tpl/images/0539/2487/3eede.png)
![\implies V[X]=\dfrac16-\left(\dfrac13\right)^2=\dfrac1{18}\implies\sqrt{V[X]}=\dfrac1{3\sqrt2}](/tpl/images/0539/2487/5573c.png)
![E[Y^2]=\displaystyle\int_{\infty}^\infty y^2f_Y(y)\,\mathrm dy=\int_0^12y^3\,\mathrm dy=\frac12](/tpl/images/0539/2487/50d59.png)
![\implies V[Y]=\dfrac12-\left(\dfrac23\right)^2=\dfrac1{18}\implies\sqrt{V[Y]}=\dfrac1{3\sqrt2}](/tpl/images/0539/2487/9bf9f.png)
![\operatorname{Cov}[X,Y]=E[(X-E[X])(Y-E[Y])]=E[XY]-E[X]E[Y]](/tpl/images/0539/2487/51a84.png)
![E[XY]=\displaystyle\int_{-\infty}^\infty\int_{-\infty}^\infty xy\,f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=\int_0^1\int_0^y2xy\,\mathrm dx\,\mathrm dy=\frac14](/tpl/images/0539/2487/317f6.png)
![\operatorname{Cov}[X,Y]=\dfrac14-\dfrac13\dfrac23=\dfrac1{36}](/tpl/images/0539/2487/dc9ba.png)
![\operatorname{Corr}[X,Y]=\dfrac{\operatorname{Cov}[X,Y]}{\sqrt{V[X]}\sqrt{V[Y]}}=\dfrac{\frac1{36}}{\left(\frac1{3\sqrt2}\right)^2}=\dfrac12](/tpl/images/0539/2487/64ada.png)
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4.52 Random variables X and Y follow a joint distribution f(x, y) = 2, 0 < x ≤ y < 1, 0, other...
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