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
Continuous
{
{
}
-
{
{ }
∑
{ }
}
∫
[
}
Problems
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?
Solution.
As shown in the figure, the answer is
.
Remark.
The Choice of sample space is relative. As long as a set is the range of some random variable, it is a
sample space in effect, although it might also be serving as the state space of another random variable.
An example is the sample space 1 above. Further in this case, the composite function of random
variable 1 and 2 above becomes a new random variable sending elements directly from the state space
to sample space 2. Also note that the probability distribution on each of the three spaces above must
be “consistent.” This consistency makes it possible to define a random variable between any pairs from
the three sets.
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.
Solution.
I. As show in the figure (there we use 0 to denote 10 for neatness), the random variable is sending
elements from the state space {12345, 12346, ……, 67890} to the sample space {2,3,4,5}.
The P.M.F. on the sample space is tallied as the following:
=
2
3
4
5
( )( )
( )( )
( )( )
( )( )
(
)
(
)
(
)
(
)
II.
Only 1-6 can possibly be the minimum number in any 5 labels drawn. {1,2,…, 6} will be the sample
space.
1
2
3
4
5
6
( )
( )
( )
( )
( )
( )
(
(
(
(
(
(
)
)
)
)
)
)
III.
Only 5-10 can possibly be the maximum number in any 5 labels drawn. {5,6,7,8,9,10} will be the
sample space. This case is symmetric to II.
10
9
8
7
6
5
( )
( )
( )
( )
( )
( )
(
(
(
(
(
(
)
)
)
)
)
)
IV.
If you draw 4 balls there must be at least 1 white since there are only 3 blacks. If you draw 3 balls it is
possible that all the 3 are blacks. Therefore the sample space is the set {1,2,3,4}.
1
2
3
4
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
such that
and
[
.
, if
( )
. Verify that such a random variable is discrete.
Solution.
First be aware of the following identity from elementary algebra:
( )
Substitute
and
by
( )
and
( )
(
)
( )
respectively will give
∑( )
This means there exists a countable set {
discrete random variable.
}
such that
{
}
. Therefore
is a
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?
Solution.
a) First,
must be
to make
. Second,
∫
b) Since
(i)
is continuous at
∫
∫
{
true:
∫
, therefore
∫
(ii)
c)
has to make ∫
}
,
∫
∫
.
.
{
}
{
Now that
,
.
}
,
∫
.
, they are 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?
Solution.
First, some clarification. A “number” here means a point of on the axis of real numbers. The usual
representation of such a point is called the “decimal expansion” of the number, using digits 0~9. For
example, the mid-point of the interval (0,1) has decimal expansion
, or equivalently,
.
a) Since the manner of drawing a point is random, each of the possible first decimal digits will be
equiprobable, making
as the answer.
b) An unconscious reasoning to this part will be “there is no difference
between a) and b) except for the superficial difference in positions of
the digits”, leading to the same
as the answer.
A more analytical way of thinking about the two questions is to
consider two random variables
{
}
{
}
to be the (deterministic) mechanisms that tell you the first and second decimal digits, respectively, of
the number (randomly) drawn.
From the graphs, it is clear the answers derived from our previous unconscious reasoning are correct.
c) Define the random variable
{
√
}
{
}
6. Find an example of two different random variables
and
with the same distribution
.
Solution.
Question 1 already provide us such an example: is the random variable 2, and is the composite of
random variables 1 and 2. Thus they share the same sample space and distribution.
Another example is to consider tossing a fair coin with sample space {
} R.V.
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
[
∫
Solution.
(a) $0.8M.
(b) [
{
}
{
}
{
}
∫
Or, from the figure we have the equation:
[
(
[
{ } )
{ }
[
{ }
∫
is defined as: