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

Cov (X,Y) = 6

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) = (1)/(12) \ $for$\ 0<x<12` 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)=(1)/(x)` for 0≤y≤x≤12

thus

`f(x,y)=(1)/(12x)`.

hence,

`E(X,Y)= \int\limits^(12)_(x=0) \int\limits^x_(y=o) xy(1)/(12x) \,dy dx`

`E(X,Y)=(!)/(24) \int\limits^(12)_(x=0) x^2 \, dx = 24`

also,

`E(Y) = \int\limits^(12)_(x=0) \int\limits^x_(y=0) y(1)/(12x) \, dydx`

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

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

= 24 - (6)(3)

= 6