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 count-able 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 ran-dom 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))
• probability density function
the probability distribution function of a continuous ran-dom 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 func-tion must equal 1
• 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:
E(X) = µ = Xx · P (X = x)
• variance (V ar(X), or σ2)
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:
V ar(X) = σ2 = X(x − µ)2 · P (X = x)
• standard deviation (SD(X), or σ)
for any discrete random variable X, the square root of its variance:
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 en-velopes 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)
Combining random variables and portfolio returns Investors build portfolios by distributing money over sev-eral investment options, but the return on each option can be viewed as a random variable (as its actual future re-turn is unpredictable); assessing the rere-turn 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 characteris-tics:
• its expected value is
E(aX + bY ) = a · E(X) + b · E(Y ) • and its variance is
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
invest-ment B, then the rate of return Rp of the portfolio is
directly related to the rates of return on the two
invest-ments, RA and RB: since
Rp = wARA + wBRB,
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)+w2B V ar(RB)
and the portfolio standard deviation is
SD(Rp) =
q
Binomial random variables • Bernoulli process
series of independent and identical trials of an experi-ment which has only two outcomes, Success and Fail-ure, 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 P (X = x) = n x pxqn−x = n! x!(n − x)!p x qn−x • binomial parameters
if X is a binomial random variable, then E(X) = µ = np
V ar(X) = σ2 = npq
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
P (X = x) = e
−µµx
x! • Poisson parameters
if X is a Poisson random variable, then E(X) = µ
Hypergeometric random variables • hypergeometric process
a sample of n individuals is randomly selected without replacement from a population of size N containing ex-actly 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 hyperge-ometric process
• hypergeometric probability mass function
where a population of N individuals contain exactly S Sucesses, we have, for x = 0, 1, . . . , n, that
P (X = x) = S x N −S n−x N n
• hypergeometric parameters
if X is a hypergeometric random variable, then
E(X) = µ = n · S N V ar(X) = σ2 = n · S N 1 − S N N − n N − 1 SD(X) = σ = s n · S N 1 − S N N − n N − 1
