A class has 10 mathematics majors, 6 computer science majors, and 4 statistics majors. Two of...

A class has 10 mathematics majors, 6 computer science majors, and 4 statistics majors. Two of these students are randomly selected to make a presentation. Let X be the number of mathematics majors and let Y be the number of computer science majors chosen.

(a) Determine the joint probability mass function p ( x , y ). This generalizes the hypergeometric distribution studied in Sect. 2.6 . Give the joint probability table showing all nine values, of which three should be 0.

(b) Determine the marginal probability mass functions by summing numerically. How could these be obtained directly? [ Hint : What type of rv is X ? Y ?]

(c) Determine the conditional probability mass function of Y given X = x for x = 0, 1, 2. Compare with the h ( y ; 2 − x , 6, 10) distribution. Intuitively, why should this work?

(d) Are X and Y independent? Explain.

(e) Determine E ( Y ∣ X = x ), x = 0, 1, 2. Do this numerically and then compare with the use of the formula for the hypergeometric mean, using the hypergeometric distribution given in part (c). (f) Determine Var( Y ∣ X = x ), x = 0, 1, 2. Do this numerically and then compare with the use of the formula for the hypergeometric variance, using the hypergeometric distribution

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orchestra answered 1 year ago

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