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Chapter 5
Random variables
• random variable
numerical variable whose value is the outcome of some
probabilistic experiment; we use uppercase letters, like
X, to denote such a variable and lowercase letters, like
x to denote the various values that X can take
• discrete vs. continuous random variables
a random variable is discrete if can only take on a countable number of distinct values, and continuous if it is
characterized by an infinite range of values within some
interval
• probability distribution function
the function that assigns probabilities to events in which
the random variable X takes on its possible values
• probability mass function
the probability distribution function of a discrete random variable, which assigns a probability to each of the
distinct values of the variable (we tabulate each value
x along with the associated probability P (X = x))
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Chapter 5
• probability density function
the probability distribution function of a continuous random variable, whose graph is a continuous curve that
describes the likelihood that X takes on values that lie
in various interval ranges
• cumulative distribution function
the function that produces values of P (X ≤ x) for each
possible value x of a (discrete or continuous) random
variable X
• properties of a probability mass function
– Since the values P (X = x) of a probability mass
function are probabilities, each must be a number
between 0 and 1
– The sum of all the values of a probability mass function must equal 1
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Chapter 5
• expected value (E(X), or µ)
for any discrete random variable X, the ideal (long-run)
average value that X takes after observing infinitely
many independent repetitions of X; computed from its
probability mass function as the sum of the products of
the values of X with their associated probabilities:
X
E(X) = µ =
x · P (X = x)
• variance (V ar(X), or σ 2)
for any discrete random variable X, the expected value
of the squared deviations from µ of the values of X;
computed from its probability mass function:
X
2
V ar(X) = σ =
(x − µ)2 · P (X = x)
• standard deviation (SD(X), or σ)
for any discrete random variable X, the square root of
its variance:
p
SD(X) = σ = V ar(X)
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Chapter 5
Expectation and risk
Uncertainty is viewed by consumers as risky; for instance,
which of these three options would you go for: (1) a coin
toss that determines which of two indistinguishable envelopes you are given, one of which contains $200 while
the other requires you to pay a $100 penalty; (2) a coin toss
that determines which of two indistinguishable envelopes
you are given, one of which contains $100 while the other is
empty; or (3) a single envelope which is known to contain
$10?
• risk loving
The risk loving consumer ignores risk and will seek the
prospect with the highest possible reward, even if it
threatens a negative expected gain (this person selects
option #1 above)
• risk neutral
The risk neutral consumer ignores risk and will accept
any prospect that offers a positive expected gain (this
person selects option #2 above)
• risk averse
The risk averse consumer expects a reward for taking a
risk (this person selects option #3 above)
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Chapter 5
Combining random variables and portfolio returns
Investors build portfolios by distributing money over several investment options, but the return on each option can
be viewed as a random variable (as its actual future return is unpredictable); assessing the return on the entire
portfolio requires understanding the joint distribution of
multiple random variables
If X and Y are two random variables, and a and b are
constants, then the variable aX + bY , called a weighted
combination of X and Y , has the following characteristics:
• its expected value is
E(aX + bY ) = a · E(X) + b · E(Y )
• and its variance is
V ar(aX+bY ) = a2 V ar(X)+2ab Cov(X, Y )+b2 V ar(Y )
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Chapter 5
Thus, if a portfolio consists of investing a fraction wA of
one’s money in investment A (wA is also called the weight
of investment A), and the remaining fraction wB in investment B, then the rate of return Rp of the portfolio is
directly related to the rates of return on the two investments, RA and RB : since
Rp = wARA + wB RB ,
we have that the expected return on the portfolio is
E(Rp) = wA · E(RA) + wB · E(RB ),
while the portfolio variance is
V ar(Rp) = wA2 V ar(RA)+2wAwB Cov(RA, RB )+wB2 V ar(RB )
and the portfolio standard deviation is
q
SD(Rp) = V ar(Rp)
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Chapter 5
Binomial random variables
• Bernoulli process
series of independent and identical trials of an experiment which has only two outcomes, Success and Failure, and for which the probability p of Success (and
therefore also the probability q = 1 − p of Failure) is
the same on each trial
• binomial random variable
counts the number of Successes in a string of n trials
of a Bernoulli process
• binomial probability mass function
For x = 0, 1, . . . , n, we have
n x n−x
n!
pxq n−x
P (X = x) =
p q
=
x!(n − x)!
x
• binomial parameters
if X is a binomial random variable, then
E(X) = µ = np
V ar(X) = σ 2 = npq
√
SD(X) = σ = npq
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Chapter 5
Poisson random variables
• Poisson process
the number of Successes of a series of independent and
identical trials of an experiment take place during an
interval of time or within a region of space so that the
probability of Success is the same in all time intervals
or spatial regions with equal duration or size
• Poisson random variable
counts the number of Successes of a Poisson process in
some time interval or spatial region
• Poisson probability mass function
where µ measures the mean number of Successes of
the Poisson process in the given time interval or spatial
region, we have, for x = 0, 1, . . . , that
e−µµx
P (X = x) =
x!
• Poisson parameters
if X is a Poisson random variable, then
E(X) = µ
V ar(X) = σ 2 = µ
√
SD(X) = σ = µ
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Chapter 5
Hypergeometric random variables
• hypergeometric process
a sample of n individuals is randomly selected without
replacement from a population of size N containing exactly S Successes, in which n is a significant fraction of
the size of N (so that distinct selections in the process
are not independent of each other, and do not have the
same probability of selecting a Success)
• hypergeometric random variable
counts the number of Successes selected in a hypergeometric process
• hypergeometric probability mass function
where a population of N individuals contain exactly S
Sucesses, we have, for x = 0, 1, . . . , n, that
S N −S
P (X = x) =
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x
n−x
N
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Chapter 5
• hypergeometric parameters
if X is a hypergeometric random variable, then
S
E(X) = µ = n ·
N
S
S N −n
V ar(X) = σ 2 = n ·
1−
N
N N −1
s
S
S N −n
SD(X) = σ = n ·
1−
N
N N −1
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