- #1

whitejac

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## Homework Statement

Consider two random variables X and Y with joint PMF given by:

P

_{XY}(k,L) = 1/(2

^{k+l}), for k,l = 1,2,3,...

A) Show that X and Y are independent and find the marginal PMFs of X and Y

B) Find P(X

^{2}+ Y

^{2}≤ 10)

## Homework Equations

P(A)∩P(B)/P(B) = P(A|B)

P(A|B) = P(A) if independent

## The Attempt at a Solution

Choosing two arbitrary numbers to show P(A|B) = P(A)

P(x<2) ∩ P(y≤1) / P(y≤1)

=P(1,1) + P(1,2) + P(1,3)... + P(1, L) ∩ P(1,1) + P(1,2) + P(1,3)...+ P(k,1) / P(y≤1)

=P(1,1) / P(1,1) + P(1,2) + P(1,3)... + P(k,1)

Note: Geometric series

=(1/4) / (1/2)(1/2^k)

=(1/4) / (1/4)(1 / (1 - (1/2)))

=(1/4) / 2

= 1/2

P(A) = P(x<2) = P(1,L) =

=1/4 (1/2

^{L})

=1/4 (1 / (1 - (1/2)))

=1/4 (2)

= 1/2

X and Y are independent.

How does one find the Marginal PMF of this equation then? The ones I've seen before in discrete sections were pre-made and finite... meaning that they were a table of results for X =1,2,3... and Y = 1,2,3... for the range of each. Should I be finding a summation for each value of X + Y?

I haven't considered part B yet.