Tutorial 1 1 01,2
1_ Prove Bonferroni’s inequality:
P(A H B) 3 P(A) —|— P(B) —1
2. Prove that
3. If n balls are distributed randomly into k urns, what is the probability that the
last urn contains j balls?
4. How many ways are there to place a indistingui
3.6 Functions ofJointIy Distributed
3.6.1
Random Variables
Sums and Quotients
Suppose that X and Y are discrete random variables taking values on the integers
and having the joint frequency function Mr. 3*). and let Z : X + Y. To ﬁnd the
frequency functio
4 Expected Values
4.1 The Expected Value of a Random Variable
4.1.1 Expectations of Functions of Random Variables
4.1.2 Expectations of Linear Combinations of Random Variables
4.2 Variance and Standard Deviation
4.2.1 A Model for Measurement Error
4.3 Cov
We saw in our discussion ot‘copulas earlier in this section that marginal densities
do not determine joint densities. For example, we can take both marginal densities
to be normal with parameters ,0: = 0 and a 2 l and use the Farlie-Morgenstern
copula wit
EXAMPLEA
Multinomial Distrilmtion
The multinomial distribution, an important generalization of the binomial distribution,
arises in the following way. Suppose that each of n independent trials can result in
one of r types of outcomes and that on each tria
If" X is a nonnegative continuous random variable. Show that
E(X) 2/ [1— F(.t‘)]d}{
(l
Apply this result to ﬁnd the mean of the exponential distribution.
"ft-n) 49¢ sfaP()(7Ix J95 X ~e3rc7§
It: Pg] «40 ) Mm“, my I-c
= (33 7‘
O X-
: fog £153 A, 4” =f”cp
3 Joint Distributions
3.1 Introduction
3.2 Discrete Random Variables
3.3 Continuous Random Variables
3.4 Independent Random Variables
3.5 Conditional Distributions
3.5.1 The Discrete Case
3.5.2 The Continuous Case
3.6 Functions of Jointly Distributed Rand
Tutorial 2 1 01.2
1. Two boys play basketball in the following way. They take turns shooting and
stop when a basket is made. Player A goes ﬁrst and has probability p. of mak-
ing a basket on any throw. Player B. who shoots second. has probability p2 of
ma
Tutorial 3 1 of 2
1 . The joint frequency function of two discrete random variables. X and Y. is given
in the following table:
X
y 1 2 3 4
1 10 .05 .02 02
2 05 .20 05 02
3 02 .05 20 04
4 02 .02 04 10
3. Find the marginal frequency functions of X and Y.
PROPOSITION C
The number of ways that :1 objects can be grouped into r classes with n,- in the
ith class,i 2 Land 222171;: 7: is
I? n!
mug-um n.!n.2!-n,.!
n n! (rt—m)! (ﬂ—Hl—ﬂg—-—H,-_l)!
ﬁlm-um. r11!(n.~n1)l(n~—m wngﬂngl ()ln.,.!
H
The numbers ( ) are c
Let T be an exponential random variable with parameter A. Let X be a discrete
random variable deﬁned as X : k if k g T < k + 1, k : 0, I, . . . . Find the
frequency function of X.
f (t) = )6“ t >0
T 1
~31:
F]. (t) = I‘ e
Let x: LT] : Ingest integer (ass '
2 Random Variables
2.1 Discrete Random Variables
2.1.1 Bernoulli Random Variables
2.1.2 The Binomial Distribution
2.1.3 The Geometric and Negative Binomial Distributions
2.1.4 The Hypergeometric Distribution
2.1.5 The Poisson Distribution
2.2 Continuous R