06:07
Probability with Applications in Engineering, Science, and Technology
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 jobability 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 $X ? ]$
(c) Determine the conditional probability mass function of $Y$ given $X=x$ for $x=0,1,2 .$ Com
pare 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 oo
the 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 of
the formula for the hypergeometric mean, using the hypergeometric distribution given in
part (c).(f) Determine $\operatorname{Var}(Y | X=x), x=0,1,2 .$ Do this numerically and then compare with the use of
the formula for the hypergeometric variance, using the hypergeometric distribution given in
part (c).