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Lecture 10

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McMaster University

Economics

ECON 2B03

Jeff Racine

Fall

Description

Lecture 10
Mean of a discrete random variable: the mean of a discrete random variable (i.e., the
average or “expected value”) is denoted by E(X), µ ,xor simply µ. The expected value is
the weighted average of all possible values of X, where the weights are the probabilities
associated with the particular X values, i.e.,
n
E(X)= ∑ xi× p(X=x ) i
i=1 ,
Where the sum is taken over all possible value of X
Expected value of a function of a random variable: Let X be a discrete random
variable, and let Y be a function of X such that Y = g(X). Then the expected value of Y
(the expected value of g(X)) is
n
E(Y)=E g([)= ] g(x )× p(X=x )
i=1 i i
Example
Consider a game of chance (i.e., “random experiment”) where it is possible to lose
$1.00, break even, win $3.00, or win $5.00 each time one plays
The probability distribution for each outcome is as follows:
Outcome [x] i -$1.00 $0.00 $3.00 $5.00
Probability 0.3 0.4 0.2 0.1
[p(xi]
The expected outcome (average, mean) for this game can be computed as follows:
Variance of a discrete random variable:
2 n 2
σ x= ∑ (xi−μ ) (=x i
i=1 Note that the variance denotes the average squared deviation from the mean, and is
2 2
referred to as E[(X−μ) ] . Using the rule E[g(X)] =E [(X)=]∧g(x) = (x−μ) ,
we obtain the formula for
The Binomial Probability Distribution
The ‘binomial’ distribution is one of the most useful discrete distributions we shall use
A binomial probability distribution shows the probabilities associated with possible
values of a discrete random variable that is generated by a type of experiment called a
Bernoulli process
A Bernoulli process consists of a sequence of n identical trials of a random experiment
such that each trial
Produces one of two possible complementary outcomes that are conventionally
called success and failure
Stands independent of any other trial so that the probability of success or failure
is constant from trial to trial
The number of successes achieved in the process is the binomial random variable
Let X denote the number of successes obtained in n independent trials in which the
probability of a success on any trial is π. The

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