Reading 4: Common Probability Distributions Flashcards
Define a probability distribution and compare and contrast discrete and continuous random variables and their probability functions.
Probability distribution: specifies the probabilities associated with the possible outcomes of a random variable. Note, there are 7 probability distributions including: uniform, binomial, normal, log normal, student t-, chi-square, and F-distribution that are used extensively in investment analysis.
Discrete random variable: can take on at most countable (possibly infinite) number of possible values. Example: the number of “yes” votes at a corporate board meeting.
Continuous random variable: In contrast, we cannot count the outcomes in a continuous random variable, we cannot describe the possible outcomes of a continuous random variable Z with a list, because the outcome which are not in the list, would always be possible. Example: rate of return on an investment.
Probability Function: specifies the probability that the random variable takes on a specific value. Discrete random variable - P(x) = P(X=x), the probability that a random variable X takes on the value x. Continuous random variable probability function is denoted f(x), and called the probability density function or just density.
Probability function has two properties, including:
1. Probability must be a number between 0 and 1 2. The sum of the probability p(x) over all values of X equals 1. If we add up the probabilities of all possible outcomes of a random variable, that sum must equal 1.
Calculate and interpret probability for a random variable given its cumulative distribution function.
Cumulative distribution function: gives the probability that a random variable X is less than or equal to a particular value x. For both continuous and discrete random variables, the shorthand notions is F(X) = P(X=x). To find F(X), we sum up the values of the probability function for all outcomes less than or equal to x. The CDF is parallel to that of the cumulative relative frequency.
Describe the properties of a discrete uniform random variable and calculate and interpret probabilities given the discrete uniform distribution function.
The discrete uniform distribution is the simplest of all probability distributions. The distributions has finite number of specified outcomes, and each outcome is equally likely.
Describe the properties of the continuous uniform distribution.
The continuous uniform distribution is the simplest continuous probability distribution. The uniform distribution has two main uses, as the basis of techniques for generating random numbers, the uniform distribution plays a role in Monte Carlo simulation. As the probability distribution that describes equally likely outcomes, the uniform distribution is an appropriate probability model to represent a particular kind of uncertainty in beliefs in which all outcomes appear equally likely.
