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A class has 10 mathematics majors, 6 computer science majors, and 4 statistics majors. Two ofthese students are randomly selected to make a presentation. Let $X$ be the number of mathematicsmajors and let $Y$ be the number of computer science majors chosen.(a) Determine the joint probability mass function $p(x, y) .$ This generalizes the hypergeometricdistribution studied in Sect. $2.6 .$ Give the jobability table showing all nine values, ofwhich 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 $X ? ]$(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.Determine $E(Y | X=x), x=0,1,2 .$ Do this numerically and then compare with the use oothe formula for the hyper geometric mean, using the hyper geometric distribution given in(e) Determine $E(Y | X=x), x=0,1,2 .$ Do this numerically and then compare with the use ofthe formula for the hypergeometric mean, using the hypergeometric distribution given inpart (c).(f) Determine $\operatorname{Var}(Y | X=x), x=0,1,2 .$ Do this numerically and then compare with the use ofthe formula for the hypergeometric variance, using the hypergeometric distribution given inpart (c).

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

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