Question

let x and y be independent uniform random variables over (0,1). find and sketch the pdf of Z=XY.

Answer 1

ANSWER::

Let X and Y are two uniform random variate over ( 0 , 1)

Therefore, the pdf of X and Y is given by,

f (x) = 1 : 0<x<1

      = 0   : Otherwise

Simillarly,

f (y) = 1 ; 0<y<1

       = 0 ; Oherwise

Since , X and Y are independent.

Therefore, the joint pdf of X and Y is given by

f (x,y) = f (x) f (y)

         = 1 ; 0<x<1 , 0<y<1

Let U = XY And V = X

Implies that X = V and Y = U/X = U/V

Therefore , the Jacobian J is diven by

J = - (1 / v)

Let x = 0 then v = 0 and x = 1 yhen v = 1

Also, y = 0 then u = 0 and y = 1 then u = v

Therefore, the new range of u and v are

0<u<v and 0<v<1

That is   

Therefore , the joint pdf of U and V are

               

                

Therefore we want to find the pdf of U=XY

Hence , to ind the marginal pdf of U

Thereore,

            

                       

                       

                      

                      

Hence , the pdf of U=XY is

                     

                                 

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