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.