Partial Solution to HW Set 7
----------------------------

These partial solutions show how to get the bounding functions
for Accept-Reject.

>>>>>>>>>>>>>>>>>>>

(a) A random vector (X_i,Y_i) with joint density f(x,y) on the positive 
quadrant equal to   K xy/(1+x^2+y^2)^5   for x,y>0  and equal to 0 for 
other x,y.  Here  K  is a constant > 0 for you to determine.


## Part (a): in polar coordinates, find theta is uniform in (0, pi/2)
##   and independent of  r  , which has density   r^3/(1+r^2)^5


> optimize(function(x) x^3/(1+x^2)^5,c(0,10), maximum=T)
$maximum
[1] 0.6546413

$objective
[1] 0.0471547

> curve(x^3/(1+x^2)^5,0,3)
> curve(x^3*exp(-4.2*x^2),0,.62,add=T, col="red")
> optimize(function(x) (1+x^2)^(-5)*exp(4.2*x^2),c(0,.62),maximum=T)
$maximum
[1] 6.631725e-05

$objective
[1] 1





> C1 = 1/integrate(function(x) (1+abs(x))*exp(-x^2/2),-1,1)$val
> C1
[1] 0.4002902
> C1*(2-2*exp(-.5)+sqrt(2*pi)*(2*pnorm(1)-1))
[1] 1

##---------------------

(b) A random variable X_i with density f(x) = C*(1+abs(x))*exp(-x^2)
for  -1 < x < 1  and equal to 0 elsewhere, where  C  is a constant for 
you to determine.

### Part (b)

> curve((1+abs(x))*exp(-x^2/2),0,1)
> optimize(function(x) (1+x)*exp(-x^2/2),c(0,1), maximum=T)
$maximum
[1] 0.618034

$objective
[1] 1.336733

> curve((1+x)*exp(-x^2/2),0,1)

> .33674/.61803^2
[1] 0.8816081
> curve(1.337-.881*(x-.61803)^2,0,1,add=T,col="red")

> optimize(function(x) (1+x)*exp(-x^2/2)/
          (1.34-.881*(x-.61803)^2),c(0,1),maximum=T)
$maximum
[1] 0.618126

$objective
[1] 0.9975622