CFA 9: Common Probability Distributions Flashcards
discrete random variable
Discrete Random Variables
A random variable that can take on at most a countable number of possible values.
continuous random variable
Discrete Random Variables
A random variable for which the range of possible outcomes is the real line (all real numbers between -infinity and +infinity) or some subset of the real line).
probability function
Discrete Random Variables
A function that specifies the probability that the random variable takes on a specific value.
probability density function
Discrete Random Variables
A function with non-negative values such that probability can be described by areas under the curve graphing the function.
cumulative distribution function
Discrete Random Variables
A function giving the probability that a random variable is less than or equal to a specified value.
Bernoulli random variable
Discrete Random Variables
A random variable having the outcomes 0 and 1.
Bernoulli trial
Discrete Random Variables
An experiment that can produce one of two outcomes.
If we let Y equal 1 when the outcome is success and Y equal 0 when the outcome is failure, then the probability function of the Bernoulli random variable Y is:
p(1) = P(Y=1) = p p(0) = P(Y=0) = 1- p
where p is the probability that the trial is a success.
binomial random variable
Discrete Random Variables
Binomial random variable X is defined as the number of successes in n Bernoulli trials.
A binomial random variable is the sum of Bernoulli random variables Yi, i =1,2, …, n:
X = Y1 + Y2 + … Yn
Where Yi is the outcome on the ith trial (w if a success, 0 if a failure). We know that a Bernoulli random variable is defined by the parameter p. The number of trials, n, is the second parameter of a binomial random variable. The binomial distribution makes these assumptions:
- The probability, p, of success is constant for all trials.
- The trials are independent.
The second assumption has great simplifying force. If indivdual trials were correlated, calculating the porbability of a given number of successes in n trials would be much more complicated.
Under the above two assumptions, a binomial random variable is completely described by two paramenters, n and p. Written as:
X ~ B(n,p)
which we read as “X has a binomial distribution with parameters n and p.” You can see that a Bernoulli random variable is a binomial random variable with n=1: Y~B(1,p).
binomial model
Discrete Random Variables
A model for pricing options in which the underlying price can move to only one of two possible new prices.
up transition probability
Discrete Random Variables
The probability that an asset’s value moves up.
down transition probability
Discrete Random Variables
The probability that an asset’s value moves down in a model of asset price dynamics.
binomial tree
Discrete Random Variables
The graphical representation of a model of asset price dynamics in which, at each period, the asset moves up with probability p or down with probability (1-p)
univariate distribution
Continuous Random Variables
A distribution that specifies the probabilities for a single random variable.
multivariate distribution
Continous Random Variables
A probability distribution that specifies the probabilities for a group of related random variables.
multivariate normal distribution
Continuous Random Variables
A probability distribution for a group of random variables that is completely defined by the means and variances of the variables plus all the correlations between pairs of the variables.
A multivariate normal distribution for the returns on n stocks is completely defined by three lists of parameters:
- The list of the mean returns on the individual securities (n means in total);
- The list of the securities’ variances of return (n variances in total); and
- The list of all the distinct pairwise return correlations: n(n-1)/2 distinct correlations in total.
standard normal distribution
Continuous Random Variables
The normal density with mean (u) equal to 0 and standard deviation equal to 1.
standardizing a random variable
Continuous Random Variables
A transformation that involves subtracting the mean and dividing the result by the standard deviation.
mean-variance analysis
Continuous Random Variables
In economic theory, mean-variance analysis holds exactly when investors are risk averse; when they choose investments so as to maximize expected utility, or satisfaction; and when either
1) returns are normally distributed, or
2) investors have quadratic utility functions.
safety-first rules
Continuous Random Variables
Rules for portfolio selection that focus on the risk that portfolio value will fall below some minimum acceptable level over some time horizon.
shortfall risk
Continuous Random Variables
The risk that portfolio value will fall below some minimum acceptable level over some time horizon.
stress testing/ scenario analysis
Continuous Random Variables
A set of techniques for estimating losses in extremely unfavorable combinations of events or scenarios.
Value at Risk (VAR)
Continuous Random Variables
A money measure of the minimum value of losses expected during a specified time period at a given level of probability.
price relative
Continuous Random Variables
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
Continuous Random Variables
The natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price.
independently and identically distributed (IID)
Continuous Random Variables
With respect to random variables, the property of random variables that are independent of each other but follow the identical probability distribution.
Monte Carlo simulation
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.
Asian call option
Monte Carlo Simulation
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
Monte Carlo Simulation
A complete pass through the steps of a simulation.
random number generator
Monte Carlo Simulation
Refers to an algorithm that produces uniformly distributed random numbers between 0 and 1.
random number (in computer simulation context)
Monte Carlo Simulation
An observation drawn from a uniform distribution. For other distributions, the term “random observation” is used in this context.
