Question

Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0, 12). Given X = x, Y is uniformly distributed on the interval (0, x). Determine Cov(X, Y) according to this model.

Answer 1

Answer:

Cov (X,Y) = 6

Step-by-step explanation:

hello,

Cov(X,Y) = E(XY) - E(X)E(Y)

we must  first find E(XY), E(X), and E(Y).

since X is uniformly distributed on the interval (0,12), then E(X) = 6.

next we find the joint density f(x,y) using the formula

$f(x,y) = g(y|x)f_{X}(x)$

$f_{X} (x) = \frac{1}{12} \ for \ 0$ this is because f is uniformly distributed on the the interval (0,12)

also since the conditional probability density of Y given X=x, is  uniformly distributed on the interval [0,x], then

$g(y|x)=\frac{1}{x}$  for 0≤y≤x≤12

thus

$f(x,y)=\frac{1}{12x}$.

hence,

$E(X,Y)= \int\limits^{12}_{x=0} \int\limits^x_{y=o} xy\frac{1}{12x} \,dy dx$

$E(X,Y)=\frac{!}{24} \int\limits^{12}_{x=0} x^2 \, dx = 24$

also,

$E(Y) = \int\limits^{12}_{x=0} \int\limits^x_{y=0} y\frac{1}{12x} \, dydx$

$E(Y)=\frac{1}{24}\int\limits^{12}_{x=0} {x} \, dx =3$

thus Cov(X,Y) = E(XY) - E(X)E(Y)

                      =  24 - (6)(3)

                      =     6