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Discrete Random Variables and Discrete Distributions
Discrete Random Variables and Discrete Distributions
Random Variables (§ 4.1)
Discrete Random Variables (§ 4.2)
Expectation & Variance of Discrete Random Variables
(§ 4.3 - 4.5)
The Bernoulli & Binomial Random Variables (§ 4.6)
The Poisson Random Variable (§ 4.7)
The Geometric Random Variable (§ 4.8.1)
The Negative Binomial Random Variable (§ 4.8.2)
The Hypergeometric Random Variable (§ 4.8.3)
Properties of Cumulative Distribution Function (§ 4.9)
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Random Variables
♦ Random Variables
A variable X , which is used to represent real numbers, are some
values determined by the outcomes of a random experiment.
Random Variable
A random variable, denoted by X , is a finite real-valued function defined on the
sample space S of a random experiment for which the probability function is
defined, and for every real number x, the set {s ∈ S : X(s) ≤ x} is an event.
Note 3.1: In mathematics, a random variable X is a function
which maps the sample space S onto a set X ⊂ R. The set X is
called the support of the random variable X .
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Random Variables
♦ Random Variables
Example 3.1
Consider an experiment by tossing a coin twice, and let
H
=
the outcome of a head;
T
=
the outcome of a tail.
Then,
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Random Variables
♦ Random Variables
Notice that
We usually denote random variables by capital letters X , Y , Z , etc.
The numeric values observed are values of a random variable (i.e.,
X = 0, 1, 2) which denote by the corresponding small letters x, y , z, etc.
Since the value of a random variable is determined by the
outcome of the experiment, we may assign probabilities to the
possible values of the random variable.
Example 3.2
Given an experiment in Example 3.1, and we consider here the probabilities of a
head and a tail on tossing a coin twice are 1/3, and 2/3, respectively. Then,
X can take on a range of values with certain probabilities while x
is an outcome from a given experiment.
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Random Variables
♦ Random Variables
Definition
The cumulative distribution function (cdf) of a random variable X , denoted by
F (x), is defined by
F (x) = P X ≤ x , −∞ < x < ∞.
Notice that
We usually refer “probability distribution function” to “cumulative distribution
function”.
F (x) denotes the probability that the random variable X takes on a value
less than or equal to x.
Example 3.3
For the random variable X given in Example 3.2, compute F (−1), F (0), F ( 12 ),
F (1) and F (2).
Solution:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Random Variables
♦ Random Variables
Properties of F (x)
1. F is a nondecreasing function; that is, if a < b, then F (a) ≤ F (b).
lim F (b) = 1.
2.
b→∞
3.
lim F (b) = 0.
b→−∞
4. F is right continuous. That is, given any b and any decreasing converged
sequence such that bn → b for n ≥ 1, we have
lim F (bn ) = F (b).
n→∞
Proof:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Random Variables
♦ Random Variables
Note 3.1:
(1) All probability questions about the random variable X can be answered in
terms of the cdf, such that
P a < X ≤ b = F (b) − F (a),
∀ a < b.
(2) The probability that the random variable X is strictly less than b is
1
P X < b = lim F b −
.
n→∞
n
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Random Variables
♦ Random Variables
Example 3.4
The distribution of a random variable X is given by

0,
x < 0;



 x/3, 0 ≤ x < 2;
2/3, 2 ≤ x < 3;
F (x) =



 5/6, 3 ≤ x < 4;
1,
x ≥ 4.
Then, find the following probabilities
(1)
(2)
(3)
(4)
P{3 < X ≤ 5}.
P{X > 1}.
P{X < 4}.
P{X = 1}.
(5) P{X = 3}.
Solution:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Discrete Random Variables
♦ Discrete Random Variables
Discrete Random Variables
A random variable X is said to be a discrete random variable if it is defined over a
sample space having a finite or a countably infinite number of sample points.
Example 3.5
(1) If X can assume only the values of 0 and 1, then X is a discrete random
variable.
(2) If X can assume only the values of 0, 1, 2, . . ., then X is a discrete random
variable.
For discrete random variables, we are interested in the events of
X = x and X ≤ x.
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Discrete Random Variables
♦ Discrete Random Variables
Probability Mass Function
For a discrete random variable X taking on values in the sample space S, the
probability mass function (pmf) of X is denoted by p(x), and is defined to be
p(x) = P{X = x}
for each x ∈ X .
Note 3.2:
1. The probability mass function p(x) is positive at most a
countable number of values of x.
2.
∑
p(x) = 1
x∈X
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Discrete Random Variables
♦ Discrete Random Variables
Graph of the pmf of X
When the support contains only a finite number of points, the pmf can be
presented by a graphical format.
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Discrete Random Variables
♦ Discrete Random Variables
Example 3.6
For the random variable X given in Example 3.1, we have
P{X = 0} = P{X = 1} =
4
9
and
P{X = 2} =
1
.
9
Note 3.3: If X is a discrete random variable taking on the values
of x1 < x2 < · · · < xn , then the probability mass function of X can
be determined by
p(x) = F (xi ) − F (xi−1 ), i = 1, . . . , n,
where F (x0 ) = 0.
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Expectation of Discrete Random Variables
If X is a discrete random variable having a probability mass function p(x), then
the expected value (or expectation) of X is denoted by E(X ), and is defined by
E(X ) =
∑
xp(x).
x∈X
Note 3.4: E(X ) is a weighted average of the possible values that
X can take on, each value being weighted by the probability that
X assumes that value.
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Example 3.7
Given the discrete random variable X in Example 3.1 and the corresponding
probability mass function in Example 3.6, compute the expected value of X .
Solution:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Expectation of A Function of A Discrete Random Variable
Let p(x) denote the probability mass function of the discrete random
variable X , and given a function of X , say g(X ).
Let Y = g(X ), then Y is also a discrete random variable.
We then use p(x) to find the probability mass function of Y , say pY (y ).
Finally, from the definition of Expected Value of a discrete random variable,
we have
E g(X ) = E(Y ) =
∑ ypY (y ).
y :pY (y )>0
Example 3.8
Given the discrete random variable X in Example 3.1 and the corresponding
probability mass function in Example 3.6. If Y = g(X ) = X 2 , then E g(X ) = 89 .
Solution:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Proposition 3.1
If X is a discrete random variable with probability mass function p(x) on a support
X , then for any real-valued function g,
E g(X ) = ∑ g(x)p(x).
x∈X
Proof:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Corollary 3.1
If X is a discrete random variable with probability mass function p(x) on a support
X , and that a and b are real constants, then
E(aX + b) = aE(X ) + b.
Proof:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Moment of a Random Variable
We define the quantity E(X n ), n ≥ 1 as the nth moment of a random variable X .
Note 3.5: For a discrete random variable X with the probability
mass function p(x), we have
E(X n ) =
∑
x n p(x).
x:p(x)>0
Variance
If X is a discrete random variable with mean µ = E(X ), then the variance of X is
defined by
Var(X ) = E (X − µ)2 .
Var(X ) ≥ 0.
Var(X ) = 0 only when the random variable X takes on
a single value.
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Note 3.6: An alternative formula for Var(X ) is
2
Var(X ) = E(X 2 ) − E(X ) .
Proof:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Example 3.9
Given the discrete random variable X in Example 3.1 and the corresponding
probability mass function in Example 3.6 whose E(X ) = 23 , E(X 2 ) = 98 . Find
Var(X ).
Solution:
Qihao Xie
Introduction to Probability and Basic Statistical Inference
Discrete Random Variables and Distributions
⇒
Expectation & Variance of Discrete Random Variables
♦ Expectation & Variance of Discrete Random Variables
Corollary 3.2
If X is a discrete random variable with probability mass function p(x) on a support
X , and that a and b are real constants, then
Var(aX + b) = a2 Var(X ).
Proof:
Standard Deviation
The square root of Var(X ) is called the standard deviation of X , and is denoted by
SD(X ) such that
p
SD(X ) = Var(X ).
Qihao Xie
Introduction to Probability and Basic Statistical Inference