Download Chapter 4. Discrete Probability Distributions

Chapter 4. Discrete Probability
Distributions
Section 4.1. Random Variables and Their
Probability Distributions
Jiaping Wang
Department of Mathematical Science
02/04/2013, Monday
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Outline
Random Variables and Probability
Functions
Distribution Functions
Examples
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Part 1. Random Variables and
Probability Functions
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Random Variable
A random variable or stochastic variable is
a variable whose value is subject to variations due to chance
(i.e. randomness, in a mathematical sense). As opposed to other
mathematical variables, a random variable conceptually does not
have a single, fixed value (even if unknown); rather, it can take
on a set of possible different values, each with an
associated probability.
For example, flip a fair coin, denote X=0
meaning tail, X=1 meaning head. So P(X=0)
= P(X=1) = ½ and P(X=n)=0 for n≠ 0 or 1 .
Then the probability
P(X≤2)=P(X=0)+P(X=1)+P(X=2)=1.
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Definition 4.1
The basic concept of "random variable" in statistics is real-valued. However, one can
consider arbitrary types such as boolean values, categorical variables, complex
numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, functions,
and processes.
Definition 4.1 A random variable is a real-valued function
whose domain is a sample space.
The random variable can be either continuous or discrete.
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Definition 4.2
A random variable X is said to be discrete if it can take on
only a finite number – or a countably infinite number –
of possible values x. The probability function of X,
denoted by p(x), assigns probability to each value x of
X so that the following conditions hold:
1. P(X=x)=p(x)≥0;
2. ∑ P(X=x) =1, where the sum is over all possible values of x.
The probability function is sometimes called the probability mass function of X to
denote the idea that a mass of probability is associated with values for discrete
points.
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Cont.
It is often convenient to list the probabilities for a discrete random
variable in a table. See following example.
x
p(x)
0
0.04
1
0.32
2
0.64
Total
1.00
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Example 4.1
A local video store periodically
puts its used movie in a bin and
offers to sell them to customers
at a reduced price. Twelve
copies of a popular movie have
just been added to the bin, but
three of these are defective. A
customer randomly selects two
of the copies for gifts. Let X be
the number of defective movies
the customer purchased. Find
the probability function of X
and graph the function.
Denote the D as the defective,
ND as the non-defective.
There are two step selections.
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Cont.
x
p(x)
0
72/132
1
54/132
2
6/132
Total
1
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Part 3. Distribution Function
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Definition 4.3
The distribution function F(b) for a random variable X is
F(b)=P(X ≤ b);
If X is discrete,
Where p(x) is the probability function.
The distribution function is often called the cumulative
distribution function (CDF).
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Cont.
x
p(x)
0
0.04
1
0.32
2
0.64
Total
1.00
P(X≤0)=0.04, P(X<0)=0,
P(X ≤1)=P(X=0)+P(X=1)=0.36 but P(X<1)=0.04
P(X ≤2)=P(X=0)+P(X=1)+P(X=2)=1.0
P(X<2)=P(X=0)+P(X=1)=0.36
P(X>2)=1.0
P(X ≤ 1.5)=P(X ≤ 1.9)=P(X ≤ 1)=0.36.
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Cont.
Note from last example, we can find F(x) is a rightcontinuous function but not left-continuous, that is
Any function satisfies the following 4 properties is a
distribution function:
1.
2.
3. The distribution function is a non-decreasing function: if a<b, then
F(a)≤ F(b). The distribution function can remain constant, but it
can’t decrease as we increase from a to b.
4. The distribution function is right-hand continuous:
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Example 4.2
A large university uses some of the
student fees to offer free use of
its health center to all students.
Let X be the number of times
that a randomly selected
student visits the center during
a semester. Based on historical
data, the distribution function of
X is given as
1. Graph F.
2. Verify that F is a distribution function.
3. Find the probability function associated with F.
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Cont.
x
p(x)
0
0.6-0=0.6
1
0.8-0.6=0.2
2
0.95-0.8=0.15
3
1.0-.095=0.05
1. Because F is zero for all values less than zero, so
2. As F is one for all values larger than 3,
3. As x increases, F(x) either remains constant or increases, it means F(x) is
nondecreasing.
4. There are three jumping points: 0, 1, 2 and 3, we can show for each point, F
is right-hand continuous. For example, when h 0+, F(2+h)0.95=F(2).
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