a. I assume the following definitions for covariance and correlation:
![\mathrm{Cov}[X,Y]=E[(X-E[X])(Y-E[Y])]=E[XY]-E[X]E[Y]](https://tex.z-dn.net/?f=%5Cmathrm%7BCov%7D%5BX%2CY%5D%3DE%5B%28X-E%5BX%5D%29%28Y-E%5BY%5D%29%5D%3DE%5BXY%5D-E%5BX%5DE%5BY%5D)
![\mathrm{Corr}[X,Y]=\dfrac{\mathrm{Cov}[X,Y]}{\sqrt{\mathrm{Var}[X]\mathrm{Var}[Y]}}](https://tex.z-dn.net/?f=%5Cmathrm%7BCorr%7D%5BX%2CY%5D%3D%5Cdfrac%7B%5Cmathrm%7BCov%7D%5BX%2CY%5D%7D%7B%5Csqrt%7B%5Cmathrm%7BVar%7D%5BX%5D%5Cmathrm%7BVar%7D%5BY%5D%7D%7D)
Recall that
![E[g(X,Y)]=\displaystyle\iint_{\Bbb R^2}g(x,y)f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy](https://tex.z-dn.net/?f=E%5Bg%28X%2CY%29%5D%3D%5Cdisplaystyle%5Ciint_%7B%5CBbb%20R%5E2%7Dg%28x%2Cy%29f_%7BX%2CY%7D%28x%2Cy%29%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy)
where
is the joint density, which allows us to easily compute the necessary expectations (a.k.a. first moments):
![E[XY]=\displaystyle\int_0^\infty\int_0^yxye^{-y}\,\mathrm dx\,\mathrm dy=3](https://tex.z-dn.net/?f=E%5BXY%5D%3D%5Cdisplaystyle%5Cint_0%5E%5Cinfty%5Cint_0%5Eyxye%5E%7B-y%7D%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D3)
![E[X]=\displaystyle\int_0^\infty\int_0^yxe^{-y}\,\mathrm dx\,\mathrm dy=1](https://tex.z-dn.net/?f=E%5BX%5D%3D%5Cdisplaystyle%5Cint_0%5E%5Cinfty%5Cint_0%5Eyxe%5E%7B-y%7D%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D1)
![E[Y]=\displaystyle\int_0^\infty\int_0^yye^{-y}\,\mathrm dx=2](https://tex.z-dn.net/?f=E%5BY%5D%3D%5Cdisplaystyle%5Cint_0%5E%5Cinfty%5Cint_0%5Eyye%5E%7B-y%7D%5C%2C%5Cmathrm%20dx%3D2)
Also, recall that the variance of a random variable
is defined by
![\mathrm{Var}[X]=E[(X-E[X])^2]=E[X^2]-E[X]^2](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BX%5D%3DE%5B%28X-E%5BX%5D%29%5E2%5D%3DE%5BX%5E2%5D-E%5BX%5D%5E2)
We use the previous fact to find the second moments:
![E[X^2]=\displaystyle\int_0^\infty\int_0^yx^2e^{-y}\,\mathrm dx\,\mathrm dy=2](https://tex.z-dn.net/?f=E%5BX%5E2%5D%3D%5Cdisplaystyle%5Cint_0%5E%5Cinfty%5Cint_0%5Eyx%5E2e%5E%7B-y%7D%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D2)
![E[Y^2]=\displaystyle\int_0^\infty\int_0^yy^2e^{-y}\,\mathrm dx\,\mathrm dy=6](https://tex.z-dn.net/?f=E%5BY%5E2%5D%3D%5Cdisplaystyle%5Cint_0%5E%5Cinfty%5Cint_0%5Eyy%5E2e%5E%7B-y%7D%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D6)
Then the variances are
![\mathrm{Var}[X]=2-1^2=1](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BX%5D%3D2-1%5E2%3D1)
![\mathrm{Var}[Y]=6-2^2=2](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BY%5D%3D6-2%5E2%3D2)
Putting everything together, we find the covariance to be
![\mathrm{Cov}[X,Y]=3-1\cdot2\implies\boxed{\mathrm{Cov}[X,Y]=1}](https://tex.z-dn.net/?f=%5Cmathrm%7BCov%7D%5BX%2CY%5D%3D3-1%5Ccdot2%5Cimplies%5Cboxed%7B%5Cmathrm%7BCov%7D%5BX%2CY%5D%3D1%7D)
and the correlation to be
![\mathrm{Corr}[X,Y]=\dfrac1{\sqrt{1\cdot2}}\implies\boxed{\mathrm{Corr}[X,Y]=\dfrac1{\sqrt2}}](https://tex.z-dn.net/?f=%5Cmathrm%7BCorr%7D%5BX%2CY%5D%3D%5Cdfrac1%7B%5Csqrt%7B1%5Ccdot2%7D%7D%5Cimplies%5Cboxed%7B%5Cmathrm%7BCorr%7D%5BX%2CY%5D%3D%5Cdfrac1%7B%5Csqrt2%7D%7D)
b. To find the conditional expectations, first find the conditional densities. Recall that
![f_{X,Y}=f_{X\mid Y}(x\mid y)f_Y(y)=f_{Y\mid X}(y\mid x)f_X(x)](https://tex.z-dn.net/?f=f_%7BX%2CY%7D%3Df_%7BX%5Cmid%20Y%7D%28x%5Cmid%20y%29f_Y%28y%29%3Df_%7BY%5Cmid%20X%7D%28y%5Cmid%20x%29f_X%28x%29)
where
is the conditional density of
given
, and
is the marginal density of
.
The law of total probability gives us a way to obtain the marginal densities:
![f_X(x)=\displaystyle\int_x^\infty e^{-y}\,\mathrm dy=\begin{cases}e^{-x}&\text{for }x\ge0\\0&\text{otherwise}\end{cases}](https://tex.z-dn.net/?f=f_X%28x%29%3D%5Cdisplaystyle%5Cint_x%5E%5Cinfty%20e%5E%7B-y%7D%5C%2C%5Cmathrm%20dy%3D%5Cbegin%7Bcases%7De%5E%7B-x%7D%26%5Ctext%7Bfor%20%7Dx%5Cge0%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
![f_Y(y)=\displaystyle\int_0^ye^{-y}\,\mathrm dx=\begin{cases}ye^{-y}&\text{for }y\ge0\\0&\text{otherwise}\end{cases}](https://tex.z-dn.net/?f=f_Y%28y%29%3D%5Cdisplaystyle%5Cint_0%5Eye%5E%7B-y%7D%5C%2C%5Cmathrm%20dx%3D%5Cbegin%7Bcases%7Dye%5E%7B-y%7D%26%5Ctext%7Bfor%20%7Dy%5Cge0%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
Then it follows that the conditional densities are
![f_{X\mid Y}(x\mid y)=\begin{cases}\frac1y&\text{for }0\le x](https://tex.z-dn.net/?f=f_%7BX%5Cmid%20Y%7D%28x%5Cmid%20y%29%3D%5Cbegin%7Bcases%7D%5Cfrac1y%26%5Ctext%7Bfor%20%7D0%5Cle%20x%3Cy%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
![f_{Y\mid X}(y\mid x)=\begin{cases}e^{x-y}&\text{for }0\le x](https://tex.z-dn.net/?f=f_%7BY%5Cmid%20X%7D%28y%5Cmid%20x%29%3D%5Cbegin%7Bcases%7De%5E%7Bx-y%7D%26%5Ctext%7Bfor%20%7D0%5Cle%20x%3Cy%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
Then the conditional expectations are
![E[X\mid Y=y]=\displaystyle\int_0^y\frac xy\,\mathrm dy\implies\boxed{E[X\mid Y=y]=\frac y2}](https://tex.z-dn.net/?f=E%5BX%5Cmid%20Y%3Dy%5D%3D%5Cdisplaystyle%5Cint_0%5Ey%5Cfrac%20xy%5C%2C%5Cmathrm%20dy%5Cimplies%5Cboxed%7BE%5BX%5Cmid%20Y%3Dy%5D%3D%5Cfrac%20y2%7D)
![E[Y\mid X=x]=\displaystyle\int_x^\infty ye^{x-y}\,\mathrm dy\implies\boxed{E[Y\mid X=x]=x+1}](https://tex.z-dn.net/?f=E%5BY%5Cmid%20X%3Dx%5D%3D%5Cdisplaystyle%5Cint_x%5E%5Cinfty%20ye%5E%7Bx-y%7D%5C%2C%5Cmathrm%20dy%5Cimplies%5Cboxed%7BE%5BY%5Cmid%20X%3Dx%5D%3Dx%2B1%7D)
c. I don't know which theorems are mentioned here, but it's probably safe to assume they are the laws of total expectation (LTE) and variance (LTV), which say
![E[X]=E[E[X\mid Y]]](https://tex.z-dn.net/?f=E%5BX%5D%3DE%5BE%5BX%5Cmid%20Y%5D%5D)
![\mathrm{Var}[X]=E[\mathrm{Var}[X\mid Y]]+\mathrm{Var}[E[X\mid Y]]](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BX%5D%3DE%5B%5Cmathrm%7BVar%7D%5BX%5Cmid%20Y%5D%5D%2B%5Cmathrm%7BVar%7D%5BE%5BX%5Cmid%20Y%5D%5D)
We've found that
and
, so that by the LTE,
![E[X]=E[E[X\mid Y]]=E\left[\dfrac Y2\right]\implies E[Y]=2E[X]](https://tex.z-dn.net/?f=E%5BX%5D%3DE%5BE%5BX%5Cmid%20Y%5D%5D%3DE%5Cleft%5B%5Cdfrac%20Y2%5Cright%5D%5Cimplies%20E%5BY%5D%3D2E%5BX%5D)
![E[Y]=E[E[Y\mid X]]=E[X+1]\implies E[Y]=E[X]+1](https://tex.z-dn.net/?f=E%5BY%5D%3DE%5BE%5BY%5Cmid%20X%5D%5D%3DE%5BX%2B1%5D%5Cimplies%20E%5BY%5D%3DE%5BX%5D%2B1)
![\implies2E[X]=E[X]+1\implies\boxed{E[X]=1}](https://tex.z-dn.net/?f=%5Cimplies2E%5BX%5D%3DE%5BX%5D%2B1%5Cimplies%5Cboxed%7BE%5BX%5D%3D1%7D)
Next, we have
![\mathrm{Var}[X\mid Y]=E[X^2\mid Y]-E[X\mid Y]^2=\dfrac{Y^2}3-\left(\dfrac Y2\right)^2\implies\mathrm{Var}[X\mid Y]=\dfrac{Y^2}{12}](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BX%5Cmid%20Y%5D%3DE%5BX%5E2%5Cmid%20Y%5D-E%5BX%5Cmid%20Y%5D%5E2%3D%5Cdfrac%7BY%5E2%7D3-%5Cleft%28%5Cdfrac%20Y2%5Cright%29%5E2%5Cimplies%5Cmathrm%7BVar%7D%5BX%5Cmid%20Y%5D%3D%5Cdfrac%7BY%5E2%7D%7B12%7D)
where the second moment is computed via
![E[X^2\mid Y=y]=\displaystyle\int_0^y\frac{x^2}y\,\mathrm dx=\frac{y^2}3](https://tex.z-dn.net/?f=E%5BX%5E2%5Cmid%20Y%3Dy%5D%3D%5Cdisplaystyle%5Cint_0%5Ey%5Cfrac%7Bx%5E2%7Dy%5C%2C%5Cmathrm%20dx%3D%5Cfrac%7By%5E2%7D3)
In turn, this gives
![E\left[\dfrac{Y^2}{12}\right]=\displaystyle\int_0^\infty\int_0^y\frac{y^2e^{-y}}{12}\,\mathrm dx\,\mathrm dy\implies E[\mathrm{Var}[X\mid Y]]=\frac12](https://tex.z-dn.net/?f=E%5Cleft%5B%5Cdfrac%7BY%5E2%7D%7B12%7D%5Cright%5D%3D%5Cdisplaystyle%5Cint_0%5E%5Cinfty%5Cint_0%5Ey%5Cfrac%7By%5E2e%5E%7B-y%7D%7D%7B12%7D%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%5Cimplies%20E%5B%5Cmathrm%7BVar%7D%5BX%5Cmid%20Y%5D%5D%3D%5Cfrac12)
![\mathrm{Var}[E[X\mid Y]]=\mathrm{Var}\left[\dfrac Y2\right]=\dfrac{\mathrm{Var}[Y]}4\implies\mathrm{Var}[E[X\mid Y]]=\dfrac12](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BE%5BX%5Cmid%20Y%5D%5D%3D%5Cmathrm%7BVar%7D%5Cleft%5B%5Cdfrac%20Y2%5Cright%5D%3D%5Cdfrac%7B%5Cmathrm%7BVar%7D%5BY%5D%7D4%5Cimplies%5Cmathrm%7BVar%7D%5BE%5BX%5Cmid%20Y%5D%5D%3D%5Cdfrac12)
![\implies\mathrm{Var}[X]=\dfrac12+\dfrac12\implies\boxed{\mathrm{Var}[X]=1}](https://tex.z-dn.net/?f=%5Cimplies%5Cmathrm%7BVar%7D%5BX%5D%3D%5Cdfrac12%2B%5Cdfrac12%5Cimplies%5Cboxed%7B%5Cmathrm%7BVar%7D%5BX%5D%3D1%7D)