Question 1
Multinomial and multivariate Hypergeometric distributions, see mostly Lecture 26, 27). David, Paul, Kate and Eva are four friends. Each of them has a favourite chocolate type. David loves white chocolate (no nuts), Paul prefers milk chocolate (no nuts), Kate’s favourite chocolate is dark (no nuts) and Eva prefer any type of chocolate that contains nuts. A mixed box of chocolate bars contains the following variety of chocolate: 10 white chocolate bars without nuts, 6 dark chocolate bars without nuts, 5 dark chocolate bars with nuts and, 7 milk chocolate bars without nuts and 2 with nuts.
(a) Kate is choosing randomly (with closed eyes) from this box 8 bars. What is the probability that within this selection there will be 2 bars for each of the boys (i.e. 2 bars for David and 2 bars for Paul), 3 chocolate bars for Eva and 1 bar for herself.
(b) Calculate the probability that out of 8 bars picked by Kate 6 will satisfy a person who is allergic to nuts (doesn’t contain nuts). (Hint: it might be easier to solve this question by using the appropriate distribution which is not multivariate).
(c) Now assume that Kate selects 8 bars one after each other but each time after selecting one bar and noticing it’s type she returns it back to the box with closed eyes and mixes the box before selecting the next bar. Under this conditions calculate the probability that the selection contains 1 bar for David, 2 bars for Paul, 3 bars for Eva and for 2 for herself
Question 2
In answering the following, you must give exact answers (fractions and surds).
(a) Determine the value of p.
(b) Copy down the table, fill in the value of p from part (a) and the marginal probabilities.
(c) Are X and Y independent? Why or why not?
(d) Now find E(X) and E(Y ).
(e) Find the probability mass function for Y conditional on X = 9 i.e. pY |X(y | 9).
(f ) Calculate E(Y |X = 9).
(g) Calculate Cov(X, Y ) (see Lecture 29).
Let Z =1√X
(h) Calculate E(Z) (Note: E(g(Y )) = Pyig(yi)pY (y)).
(i) Calculate Var(Z).
(j) Calculate Cov(Y, Z) (Note: E(g(X, Y )) = PxiPyig(xi, yi)pXY (xi, yi
Question 3.
(See mostly Lecture 27, 28 and 29, and prac 10.) Consider the joint probability density function for random variables X and Y .
fXY (x, y) = (2 − x)24+23y3 0 ≤ x ≤ 2, 0 ≤ y ≤ 10 otherwise
(a) Find the cumulative distribution function FXY (x, y) for the region where x ∈ [0, 2],y ∈ [0, 1] (see Lecture 27).
(b) Calculate PX < 1, Y > 112
(c) Find the marginal distribution fX(x).
(d) Find the marginal distribution fY (y).
(e) Are X and Y independent? Why or why not?
(f ) Calculate E(X) and E(Y ).
(g) Calculate Cov(X, Y ).
(h) Calculate Cov
4X + 1,12Y
