7. Conditioning on a rv; Independence of r.v.'s

Joint PMF

Conditional PMF

Marginal PMF

By keeping y fixed

In PMF Notation

Joint PMF factors out as a product of the marginal PMFs of the two random variable

Entry of the joint PMF is equal to the product of the corresponding entries of the marginal PMFs. Therefore the two random variable X and Y are independent in the conditional universe.