(a) looks right, but you need to place () before the factorials and cancel that common term. This is a multinomial.
(b) does not make sense mathematically. You can't have y in a function that is only a function on x. This is just a binomial.
In a smoking survey among boys between the ages of 12 and 17, 78% prefer to date
nonsmokers. 1% prefer to date smokers, and 21% don’t care. Suppose seven such boys are
selected randomly and X equals the number who prefer to date nonsmokers and Y equals the
number who prefer to date smokers.
(a) Determine the joint p.m.f. of X and Y. Be sure to include the support of the p.m.f.
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(b) Find the marginal p.m.f. of X. Again, include the support.
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Thank you!