Chapter 2
Probability
is the measure of ones belief that a future (random)
:
event will occur.
A random
event is one whose occurrence cannot be predicted
:
with certainty.
However, we can often understand the long-run :
relative
frequency (proportion)
Chapter 3. Discrete random variables and probability
distributions.
Defn:
A :
random:
variable (r.v.) is a function that takes a sample point
:
in S and maps it to it a real number.
That is, a r.v. Y maps the sample space (its domain) to the real
number l
Chapter 6: Functions of Random Variables
We are often interested in a function of one or several random
variables, U(Y1 , . . . , Yn ).
We will study three methods for determining the distribution of a
function of a random variable:
1. The method of cdfs
Multivariate Distributions
In many applications we measure several r.v.s per individual.
Examples:
:
For a set of households, we can measure both income and the
number of children.
For a set of giraffes, we can measure their height and weight.
We can m
Continuous random variables
Continuous r.v.s take an uncountably infinite number of possible
values.
Examples:
:
Heights of people
Weights of apples
Diameters of bolts
Life lengths of light-bulbs
We cannot assign positive probabilities to every partic
Math 447
Final Exam
Fall 2015
No books, no notes, only SOA-approved calculators. Please put your answers in the spaces provided!
Section:
Name:
Question Points Score
1
8
2
6
3
10
4
19
5
9
6
10
7
14
8
14
9
23
10
14
Total:
127
1. (8 points) (Monty Hall, Aga
Math 447
Test 2 Solutions
1. Let the random variable Y have distribution function
0,
y
,
8
F (y) = y2
16 ,
1,
Fall 2015
y 0,
0 < y < 2,
2 y < 4,
y 4.
(a) (5 points) What is the probability density function of
in the denition of F !)
Solution. This is ex