Probability Applications Flashcards
Discrete random variable
A random variable that can take on at most a countable number of possible values.
Random Variable
a quantity whose future outcomes are uncertain
Continuous random variable
A random variable for which the range of possible outcomes is the real line (all real numbers between −∞ and +∞ or some subset of the real line).
Probability function
A function that specifies the probability that the random variable takes on a specific value.
Probability density function
A function with non-negative values such that probability can be described by areas under the curve graphing the function.
Cumulative distribution function
A function giving the probability that a random variable is less than or equal to a specified value.
Bernoulli random variable
A random variable having the outcomes 0 and 1.
n = 1: Y ~ B(1, p).
Binomial random variable
The number of successes in n Bernoulli trials for which the probability of success is constant for all trials and the trials are independent.
a binomial random variable is completely described by two parameters, n and p. We write
X ~ B(n,p)
Probability function for a binomial random variable
x n-x
=n!/(n−x)!x! [p (1−p) ]
Variance of Bernoulli random variable
p(1 − p)
Variance of binomial random variable
np(1 − p)
Mean of a continuous random variable
μ = (a + b)/2
Variance of a continuous random variable
σ2 = (b − a)2/12.
Defining characteristics of a normal distribution
The normal distribution is completely described by two parameters—its mean, μ, and variance, σ2. We indicate this as X ~ N(μ, σ2) (read “X follows a normal distribution with mean μ and variance σ2”). We can also define a normal distribution in terms of the mean and the standard deviation, σ (this is often convenient because σ is measured in the same units as X and μ). As a consequence, we can answer any probability question about a normal random variable if we know its mean and variance (or standard deviation).
The normal distribution has a skewness of 0 (it is symmetric). The normal distribution has a kurtosis (measure of peakedness) of 3; its excess kurtosis (kurtosis − 3.0) equals 0.17 As a consequence of symmetry, the mean, median, and the mode are all equal for a normal random variable.
A linear combination of two or more normal random variables is also normally distributed.
Normal density function
f(x)=( 1/σ√2π ) exp( −(x−μ)2 / 2σ2 ) for −∞<+ ∞
Standard normal distribution
The normal density with mean (μ) equal to 0 and standard deviation (σ) equal to 1.
Standardization formula
Z = (X – μ)/σ
A transformation that involves subtracting the mean and dividing the result by the standard deviation.
Value at Risk (VAR)
A money measure of the minimum value of losses expected during a specified time period at a given level of probability.
Scenario analysis
Analysis that shows the changes in key financial quantities that result from given (economic) events, such as the loss of customers, the loss of a supply source, or a catastrophic event; a risk management technique involving examination of the performance of a portfolio under specified situations. Closely related to stress testing.
Stress testing
A set of techniques for estimating losses in extremely unfavorable combinations of events or scenarios.
Monte Carlo simulation
An approach to estimating a probability distribution of outcomes to examine what might happen if particular risks are faced. This method is widely used in the sciences as well as in business to study a variety of problems.
Monte Carlo simulation involves the use of a computer to represent the operation of a complex financial system. A characteristic feature of Monte Carlo simulation is the generation of a large number of random samples from specified probability distribution(s) to represent the operation of risk in the system. Monte Carlo simulation is used in planning, in financial risk management, and in valuing complex securities. Monte Carlo simulation is a complement to analytical methods but provides only statistical estimates, not exact results.
Asian call option
A European-style option with a value at maturity equal to the difference between the stock price at maturity and the average stock price during the life of the option, or $0, whichever is greater.
Simulation trial
A complete pass through the steps of a simulation.
Historical simulation
Another term for the historical method of estimating VAR. This term is somewhat misleading in that the method involves not a simulation of the past but rather what actually happened in the past, sometimes adjusted to reflect the fact that a different portfolio may have existed in the past than is planned for the future.
Historical simulation is an established alternative to Monte Carlo simulation that in one implementation involves repeated sampling from a historical data series. Historical simulation is grounded in actual data but can reflect only risks represented in the sample historical data. Compared with Monte Carlo simulation, historical simulation does not lend itself to “what if” analyses.
Relationship between normal and lognormal distributions
If continuously compounded returns are normally distributed, asset prices are lognormally distributed. This relationship is used to move back and forth between the distributions for return and price. Because of the central limit theorem, continuously compounded returns need not be normally distributed for asset prices to be reasonably well described by a lognormal distribution.
Roy’s safety first criterion
Roy’s safety-first criterion, addressing shortfall risk, asserts that the optimal portfolio is the one that minimizes the probability that portfolio return falls below a threshold level. According to Roy’s safety-first criterion, if returns are normally distributed, the safety-first optimal portfolio P is the one that maximizes the quantity [E(RP) − RL]/σP, where RL is the minimum acceptable level of return.
Standard normal random variable
The standard normal random variable, denoted Z, has a mean equal to 0 and variance equal to 1. All questions about any normal random variable can be answered by referring to the cumulative distribution function of a standard normal random variable, denoted N(x) or N(z).
Normal random variable
For a normal random variable, approximately 68 percent of all possible outcomes are within a one standard deviation interval about the mean, approximately 95 percent are within a two standard deviation interval about the mean, and approximately 99 percent are within a three standard deviation interval about the mean.
A normal random variable, X, is standardized using the expression Z = (X − μ)/σ, where μ and σ are the mean and standard deviation of X. Generally, we use the sample mean X⎯⎯⎯ as an estimate of μ and the sample standard deviation s as an estimate of σ in this expression.
Normal distribution
The normal distribution is a continuous symmetric probability distribution that is completely described by two parameters: its mean, μ, and its variance, σ2.
Univariate and multivariate distribution
A univariate distribution specifies the probabilities for a single random variable. A multivariate distribution specifies the probabilities for a group of related random variables.
Normal distribution standard deviation intervals
Approximately 50 percent of all observations fall in the interval μ ± (2/3) σ.
Approximately 68 percent of all observations fall in the interval μ ± σ.
Approximately 95 percent of all observations fall in the interval μ ± 2σ.
Approximately 99 percent of all observations fall in the interval μ ± 3σ.
Frequently referenced values in the standard normal table
The 90th percentile point is 1.282: P(Z ≤ 1.282) = N(1.282) = 0.90 or 90 percent, and 10 percent of values remain in the right tail.
The 95th percentile point is 1.65: P(Z ≤ 1.65) = N(1.65) = 0.95 or 95 percent, and 5 percent of values remain in the right tail. Note the difference between the use of a percentile point when dealing with one tail rather than two tails. Earlier, we used 1.65 standard deviations for the 90 percent confidence interval, where 5 percent of values lie outside that interval on each of the two sides. Here we use 1.65 because we are concerned with the 5 percent of values that lie only on one side, the right tail.
The 99th percentile point is 2.327: P(Z ≤ 2.327) = N(2.327) = 0.99 or 99 percent, and 1 percent of values remain in the right tail.
Negative numbers in the standard normal table
For a non-negative number x, use N(x) from the table. Note that for the probability to the right of x, we have P(Z ≥ x) = 1.0 − N(x).
For a negative number −x, N(−x) = 1.0 − N(x): Find N(x) and subtract it from 1. All the area under the normal curve to the left of x is N(x). The balance, 1.0 − N(x), is the area and probability to the right of x. By the symmetry of the normal distribution around its mean, the area and the probability to the right of x are equal to the area and the probability to the left of −x, N(−x).
For the probability to the right of −x, P(Z ≥ −x) = N(x).
Price relative
A ratio of an ending price over a beginning price; it is equal to 1 plus the holding period return on the asset.
Continuously compounded return
The natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price.
r t,t+1 = ln(St+1/St) = ln(1 + Rt,t+1)
