Question

define U=x+y, V=x-y.

find the joint and marginal pdf of U and V

Answer 1

Let X and Y be independent exponential random variables with common parameter .

Consider , U = X + Y & V = X - Y

now , according to question :

it is given that ,

f(X,Y) = 1 / * exp(-(X+Y)/)

, X>0,Y>0

Now , since U = X+Y & V = X-Y , and there is only one solution given by:

X =(U+V)/2 & Y =(U-V)/2

and also the jacobian of the transformation is given by J(X,Y) = -2

And now , f(U,V) = 1/2 * exp(-u/)

represents the joint pdf of U and V .

Now , marginal pdf of U and V is given by :

f(U) = f( U,V) dV

f(U) = 1/2 exp(-u/) dV

f(U) = u / * exp(-u/) , is the marginal pdf of U.

f(V) = f( U,V) dU

f(V) =  1/2 exp(-u/) dU

f(V) = 1/2 * exp(-|V|/) , is the marginal pdf of V .