Download Random Variable

THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1301 PROBABILITY AND STATISTICS I, FALL 2010
EXAMPLE CLASS 3
Random Variable
Elements of Theory
Random Variable as a (measurable) Function between a State Space and a Sample Space
A function (with some requirement)
random variable.
-
defined on the state space
{ } is called a
Domain is the state space
Range is usually a numbers set, e.g., or its subsets, for easy manipulation.
The range is called the sample space of the random variable. There is no intrinsic difference on the
nature between a sample space and a state space—they are just two sets with some requirement, called
“measurability.” They are just domain and range of a “function” with some requirement, called
“measurability.”
“Random Variable” is a “Function”
The variable perspective is adopted by an observer of a random experiment. The observer is only able to
observe/know/measure/obtain information based on the sample space. For the observer, all she could see is a
variable dancing (randomly) on the sample space. This is the perspective that we will primarily study in this course.
The function perspective is adopted by someone who would have a “divine” capacity in understanding (a
deterministic part of) the design of the random mechanism, in particular, her capacity in seeing the existence of an
underlying state space as the domain of a function sending elements of the domain to the state space. You will be
studying this perspective in an advanced course of probability.
Distribution: Law of A Random Variable’s Dance
There is a law governing how any random variable to be observed in its sample space. The law is a probabilistic one,
called the probability distribution of a random variable. There are two qualifications for any real-valued function
to be a probability density/mass function, aka, probability function:
1)
for any
,
or ∑
2) ∫
.
Real-valued Random Variables
Generally, we categorize all real-valued random variables in 3 groups: (1) Discrete Random Variables (its sample
space is a discrete subset in ); (2) Continuous Random Variables (its sample space is a “continuous” subset in ); (3)
Partially Discrete and Partially Continuous Random Variables. The Distribution of a Discrete/Continuous Random
Variable is called its Probability Mass/Density Function because of a superficial difference in mathematical
treatments and graphical representation.
Discrete
Random
Variable
Probability
Density
Function
Probability
Mass
Function
Cumulative
Distribution
Function
Expectation
Problems
Continuous
{
{
}
-
{
{ }
∑
{ }
}
∫
[
}
1. Suppose that you are invited to play a game with the following rules: One of the numbers 2, … , 12
is chosen at random by throwing a pair of dice and adding the numbers shown. You win 9 dollars in
case 2, 3, 11, or 12 comes out, or lose 10 dollars if the outcome is 7. Otherwise, you do not win or lose
anything. This defines a function on the set of all possible outcomes {2, … , 12}, the value of the
function being the corresponding gain (or loss if the value is negative). What is the probability that the
function takes positive values, i.e,, that you will win some money?
2. An urn contains 7 white balls numbered 1, 2, …, 7 and 3 black balls numbered 8, 9, 10. Five balls
are randomly selected without replacement.
Now give the distribution:
I.
II.
III.
IV.
of the number of white balls in the sample;
of the minimum number in the sample;
of the maximum number in the sample;
of the minimum number of balls needed for selecting a white ball.
3. A random variable is called discrete whenever there is a countable set
A random variable is said to have the binomial distribution
, where
{ }
for
( )
. Verify that such a random variable is discrete.
such that
and
[
.
, if
4. The amount of bread (in hundreds of kilos) that a bakery sells in a day is a random variable with
density
{
a) Find the value of which makes a probability density function.
b) What is the probability that the number of kilos of bread that will sold in a day is, (i) more than
300 kilos? (ii) between 150 and 450 kilos?
c) Denote by and the events in (i) and (ii), respectively. Are and independent events?
5. A number is randomly chosen from the interval (0,1). What is the probability that:
a) its first decimal digit will be a 1;
b) its second decimal digit will be a 5;
c) the first decimal digit of its square root will be a 3?
6. Find an example of two different random variables
and
with the same distribution
.
7. (a) A project will bring $1M profit if completed. If the probability of completion is 80%, what is the
expected profit?
(b) For every set
of real numbers, we define the indicator function by
{
Show that
[
∫