Common Probability Distributions Flashcards
1. Learn the difference between discrete and continuous probability distributions. 2. The binomial and normal distributions are the most important here. You must learn the properties of both distributions and memorize the formulas for the probability of a particular value when given a binomial probability distribution. 3. Learn what shortfall risk is and how to calculate and use Roy's safety-first criterion. 4. Know how to standardize a normally distributed random variable, use a z-t
Probability Distribution
A probability distribution describes the probabilities of all the possible outcomes for a random variable.
Discrete Random Variable
A discrete random variable is one for which the number of possible outcomes can be counted, and for each possible outcome, there is a measurable and positive probability.
Probability Function
A probability function, denoted p(x), specifies the probability that a random variable is equal to a specific value.
Continuous Random Variable
A continuous random variable is one for which the number of possible outcomes is infinite, even if lower and upper bounds exist.
Cumulative Distribution Function
A cumulative distribution function (cdf), or simply distribution function, defines the probability that a random variable, X, takes on a value equal to or less than a specific value, x. It represents the sum, or cumulativt valut, of the probabilities for the outcomes up to and including a specified outcome.
Discrete Uniform Variable
A discrete uniform random variable is one for which the probabilities for all possible outcomes for a discrete random variable are equal.
Binomial Random Variable
A binomial random variable may be defined as the number of “successes” in a given number of trials, where by the outcome can be either “success” or “failure.”
Bernoulli random variable
A binomial random variable for which the number of trials is 1 is called a Bernoulli random variable
Binomial Probability Function
Check formula online
The variance of a binomial random variable is given by:
variance of X = np(1 - p)
Tracking error
Tracking error is the difference between the total return on a portfolio and the total return on the benchmark against which its performance is measured.
Continuous uniform distribution
The continuom uniform distribution is defined over a range that spans between some lower limit, a, and some upper limit, b, which serve as the parameters of the distribution.
Normal Distribution Properties
The normal distribution has the following key properties:
1. It is completely described by its mean, J.L, and variance. In words, this says that “X is normally distributed with mean and variance.”
- Skewness = 0, meaning that the normal distribution is symmetric about its mean. mean = median = mode.
- Kurtosis = 3; this is a measure of how flat the distribution is. Recall that excess kurtosis is measured relative to 3, the kurtosis of the normal distribution.
- A linear combination of normally distributed random variables is also normally distributed.
- The probabilities of outcomes further above and below the mean get smaller and smaller but do not go to zero (the tails get very thin but extend infinitely).
Univariate Distribution
The distributions of a single Variable
Multivariate Distribution
A multivariate distribution specifies the probabilities associated with a group of random variables and is meaningful only when the behavior of each random variable in the group is in some way dependent upon the behavior of the others.
Standard Normal Distribution
The standard normal distribution is a normal distribution that has been standardized so that it has a mean of zero and a standard deviation of 1.
