# Change of variable transformation

#### dim&dimmer

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)

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

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