Statistics 2EA1
Random Variables
Each outcome of an experiment can be associated
with a number by specifying a rule of association.
Such a rule of association is called a random
variable — a variable because different
numerical values are possible, and random because
the observed value depends on which of the
possible experimental outcomes results.
Random Variables
Definition:
For a given sample space S of some experiment, a
random variable (rv) is any rule that associates a
number with each outcome in S.
In mathematical language, a random variable is a
function whose domain is the sample space and
whose range is the set of real numbers.
Random Variables
Random variables are customarily denoted by
uppercase letters, such as X and Y, near the end
of our alphabet.
In contrast to our previous use of a lowercase
letter, such as x, to denote a variable, we will now
use lowercase letters to represent some
particular value of the corresponding random
variable.
The notation X(s) = x means that x is the value
associated with the outcome s by the rv X.
Example 3.1
Random Variables
Definition:
Any random variable whose only possible values
are 0 and 1 is called a Bernoulli random variable.
Example 3.2
Two Types of Random Variables
Definition:
Example 3.3
Example 3.4
A discrete random variable is an rv whose possible values
either constitute a finite set or else can be listed in an
infinite sequence in which there is a first element, a
second element, and so on.
A random variable is continuous if both of the following
apply:
1.Its set of possible values consists either of all numbers in a single
interval on the number line (possibly infinite in extent, e.g., from -∞
to ∞) or all numbers in a disjoint union of such intervals (e.g., [0, 10]
union [20, 30]).
2.No possible value of the variable has positive probability, that is,
P(X = c) = 0 for any possible value c.
Probability Distributions for Discrete
Random Variables
The probability distribution for a discrete
random variable X resembles the relative
frequency distributions we constructed in
Chapter 1. It is a graph, table or formula that
gives the possible values of X and the probability
p(x) associated with each value.
We must have
p ( x)  0 and
 px   1
Probability Distributions for Discrete
Random Variables
Definition:
The probability distribution or probability mass
function (pmf) of a discrete rv is defined for
every number x by:
px  P X  x  Pall s  S : X s   x
Example 3.8
Example 3.9
Example 3.10
Probability Distributions for Discrete
Random Variables
Definition:
Suppose p(x) depends on a quantity that can be
assigned any one of a number of possible values,
with each different value determining a
different probability distribution.
Such a quantity is called a parameter of the
distribution.
The collection of all probability distributions for
different values of the parameter is called a
family of probability distributions.
Example 3.12
Probability Distributions for Discrete
Random Variables
Definition:
Suppose p(x) depends on a quantity that can be
assigned any one of a number of possible values,
with each different value determining a
different probability distribution.
Such a quantity is called a parameter of the
distribution.
The collection of all probability distributions for
different values of the parameter is called a
family of probability distributions.
Example 3.12
The Cumulative Distribution Function
Definition:
The cumulative distribution function (cdf) F(x)
of a discrete rv variable X with pmf p(x) is
defined for every number x by
F  x   P X  x  
 p y 
y: y  x
For any number x, F(x) is the probability that the
observed value of X will be at most x.
Example 3.13
The Cumulative Distribution Function
Proposition:
For any two numbers a and b with a  b
 
Pa  X  b   F b   F a 
For any two integers a and b with a  b
Pa  X  b  F b  F a 1
Example 3.15
Expected Values
Let X be a discrete rv with set of possible
values D and pmf p(x).
The expected value or mean value of X,
denoted by E(X) or  X is
E  X    X   x. p x 
xD
Provided that
 x . px   
xD
Expected Values
Expected Value E(X):
The value that you would expect to
observe on average if the experiment
is repeated over and over again
Examples
Page 110 - 111
Leave out
Examples 3.19 & 3.20
Expected Value of a Function
Proposition:
If the rv X has a set of possible values D
and pmf p(x), then the expected value of
any function h(X), denoted by E[h(X)] or  h  X 
is computed by
E h X    h x . p  x 
D
assuming that
Examples
Page 112 - 113
 h x  . p  x   
D
Expected Value of a Function
PROOF!
Proposition:
EaX  b  aE X   b
The Variance of X
Definition:
Let X have pmf p(x) and expected value  .
Then the variance of X, denoted by V(X) or  X2
is

V  X     x    . p x   E  X   
2
D
The standard deviation (SD) of X is
X  X
Example 3.24
2
2

A Shortcut Formula
PROOF!
Proposition:
V X    2


2
2
  x . px   
D

 E X
Example 3.25
2
 E  X 
2
Rules of Variance
PROOF!
Proposition:
2
2
2
V aX  b    aX

a
.

b
X
 aX b  a .  X
Example 3.26