Tutorial Exercise (Week 6)

MATH2421 T1C
1.
2.
Tutorial Exercise (Week 6)
(Chapter 4 Theoretical Exercise 32) A jar contains n chips. Suppose that a boy successively
draws a chip from the jar with replacement. The process continues until the boy draws a chip
that he has previously drawn. Let X denote the number of draws. Compute its probability mass
function.
(Chapter 4 Theoretical Exercise 34) From a set of n elements, a nonempty subset is chosen at
random such that all of the nonempty subsets are equally likely to be selected. Let X denote the
number of elements in the chosen subset.
n
n
n
 n
n
(a) Using  r    n  2 n 1 and  r 2    n(n  1)2 n2 , show that E[ X ] 
and
n 1
r 1  r 
r 1
r
1
2 
2
2 n2
n2
n2
 n(n  1)2
.
Var ( X ) 
(2 n  1) 2
(b) For large n, show that Var ( X ) 
n
. Compare this formula with the limiting form of Var(Y),
4
where P(Y = i) = 1/n, i = 1, 2, …, n.
3.
(Chapter 4 Theoretical Exercise 35) An urn initially contains 1 red ball and 1 blue ball. At each
stage, a ball is randomly chosen and then replaced with another of the same color. Let X denote
the number of selections required to get a blue ball.
(a) Find the probability mass function of X.
(b) Show that the probability that a blue ball is eventually chosen is 1.
(c) Find E[X].
4.
(Chapter 4 Theoretical Exercise 30) A subset of size n is randomly selected from {1, 2, …, N}.
Let X denote the largest number in the subset.
(a) Find the probability mass function of X.
 n n  i 1
 , find E[X].
(b) Using     
 r  i r  r  1
5.
(Chapter 4 Theoretical Exercise 31) A subset of size n is randomly selected from {1, 2, …, m +
n}. Let X denote the number of selected numbers that exceed each of those remaining. Find the
probability mass function of X.
6.
(Chapter 4 Theoretical Exercise 14) The probability that a family has n children is apn, where a
and p are constants and, n ≥ 1.
(a) What proportion of families has no children?
(b) If each child is equally likely to be a boy or a girl, what proportion of families consists of
exactly k boys?
7.
(Chapter 4 Theoretical Exercise 20) Consider n coins, each of which independently comes up
heads with probability p. Suppose that n is large and p is small so that np is moderate. Suppose
that all n coins are tossed; if at least one comes up heads, the experiment ends; if not, we again
toss all n coins, and so on. Let X denote the total number of heads that appear. Estimate the
probability mass function of X.
8.
(Chapter 4 Theoretical Exercise 23) Consider a random collection of n persons. Let Ei denote
the event that there are at least 3 birthdays on the ith day of a year, where i = 1, 2, …, 365.
(a) Find P(Ei).
(b) Estimate the probability that no 3 persons share the same birthday.
(c) Hence, estimate the smallest value of n for which the probability that at least 3 persons
share the same birthday exceeds 0.5.
9.
(Chapter 4 Theoretical Exercise 15, 17)
(a) Suppose that n independent tosses of a coin having probability p of coming up heads are
made. Show that the probability that an even number of heads results is
1
[1  (1  2 p) n ] .
2
1
(b) Hence, show that P( X is even )  [1  e 2 ] , where X ~ Poisson(λ).
2
10. (Chapter 4 Theoretical Exercise 13, 18)
(a) Let X ~ Bin(n, p). What value of p maximizes P(X = k) for some k = 0, 1, …, n?
(b) Let Y ~ Poisson(λ). What value of λ maximizes P(Y = k) for some k ≥ 0?
11. (Chapter 4 Theoretical Exercise 27)
(a) Prove that a geometric random variable is memoryless.
(b) Prove that a memoryless discrete random variable is a geometric random variable.