New compulsory home works

Home works given on November 3 (due on November 10):

1. Choose a random point X on the circumference of the circle with radius .5
   and center (0,.5) in the x-y coordinate system. Project X from the point
   (0,1) to the x-axis, and let Y be the image of X under the projection.
   Show that the distribution of Y is Cauchy.

2. RND is a uniformly distributed random variable in [0,1], which can be
   generated by the computer. What is the distribution of aRND^c where a>0
   and c<0 are constants. Give the distribution function and the density
   function.

Home work given on November 10 (due on November 17):

3. Let X be an exponentially distributed random variable with parameter 2.
   (Remember that the density funtion is then f(x)=2exp(-2x) if x>0 and 0
   otherwise.) Identify the density function of Z=exp(X).

Home work given on November 17 (due on November 24):

4. Let X and Y be two independent standard normal random variables. Consider
   the 2x2 matrix

     [ 3  0 ]
   A=[ 1  1 ]

   What is the joint density function of the random variable vector

       [ X ]
   A . [ Y ] ?

Home work given on November 24 (due on December 1):

5. Let (X1,X2) be a random variable vector with normal distribution. The
   expectation is (0,0), and the covariance matrix is

     [ 2  1 ]
   C=[ 1  1 ] .

   (a) Compute the marginal density function of the second coordinate X2.

   (b) What is the conditional density function of X1 given X2?

Home work given on December 1 (due on December 8):

6. Let 0<s<t and consider the joint distribution of (W(s),W(t)) where
   W is the Brownian motion. Show that (1/2W(4s),1/2W(4t)) has the same
   distribution as (W(s),W(t)).

   Hint: Show the equality of the densities. The vector (1/2W(4s),1/2W(4t))
   can be got by multiplying the vector (W(4s),W(4t)) by the matrix

     [1/2 0 ]
   A=[ 0 1/2].