- 22

- 0

**1. Homework Statement**

Let X ~ P (lambda_x) , Y ~ P (lambda_y) X<Y independent.

Use change of variable technique to show that,

X + Y ~ P (lambda_x + lambda_y)

Verify your result using MGFs.

**2. Homework Equations**

**3. The Attempt at a Solution**

Really struggling!

Started p_XY(x,y) = p_X(x) p_Y(y) (independent)

= {e^(-lambda_x) lambda^x}/x! . {e^(-lambda_y) lambda^y}/y! , x,y = 0, 1, 2 ,....

U = q_1(X,Y) = X + Y , V = q_2 (X,Y) = Y

X = w_1 (U,V) = U - V , Y = w_2 (U,V) = V

p_UV (u,v) = {e^(-lambda_u) lambda^u}/u! . {e^(-lambda_v) lambda^v}/v! , u,v = 0, 1, 2 ,....

I dont really know what I'm doing !! Any assisstance would be great

dim