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Statistics 2EA1
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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.
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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.
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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
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Random Variables
Definition:
Any random variable whose only possible values
are 0 and 1 is called a Bernoulli random variable.
Example 3.2
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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.
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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
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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
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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
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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
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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
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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
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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
 x . p(x )  
Provided that
xD
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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
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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
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Expected Value of a Function
PROOF!
Proposition:
EaX + b = aE ( X ) + b
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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
2

D
The standard deviation (SD) of X is
X = X2
Example 3.24
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A Shortcut Formula
PROOF!
Proposition:
V (X ) =  2


=  x 2 . p( x ) −  2
D

( )
= E X 2 − E ( X )
2
Example 3.25
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Rules of Variance
PROOF!
Proposition:
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2
2
V (aX + b ) =  aX
+b = a .  X
 aX +b = a .  X
Example 3.26
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