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whitejac
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Homework Statement
Consider two random variables X and Y with joint PMF given by:
PXY(k,L) = 1/(2k+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(X2 + Y2 ≤ 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 (1/2L)
=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.