msk2222's version from 2018-02-18 17:35

## Summary

INTRODUCTION TO COMMON PROBABILITY DISTRIBUTIONS-DISCRETE RANDOM VARIABLES

## The Discrete Uniform Distribution

definitionThe distribution has a finite number of specified outcomes, and each outcome is equally likely.
The cdf has two other characteristic propertiesThe cdf lies between 0 and 1 for any x: 0 ≤ F(x) ≤ 1. As we increase x, the cdf either increases or remains constant.
The Binomial Distribution When we make probability statements about a record of successes and failures, or about anything with binary outcomes, we often use the binomial distribution
binomial random variableX 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:

Now we extend the model to describe stock price movement on three consecutive days. Each day is an independent trial. The stock moves up with constant probability p (the up transition probability); if it moves up, u is 1 plus the rate of return for an up move. The stock moves down with constant probability 1 − p (the down transition probability); if it moves down, d is 1 plus the rate of return for a down move. We graph stock price movement in Figure 2, where we now associate each of the n = 3 stock price moves with time indexed by t. The shape of the graph suggests why it is a called a binomial tree. Each boxed value from which successive moves or outcomes branch in the tree is called a node; in this example, a node is potential value for the stock price at a specified time.

## CONTINUOUS RANDOM VARIABLES

pdf for a uniform random variable isf(x) = 1/(b-a) for x between a and b & f(x)= 0 otherwise
middle line of the expression for the cdf captures this relationshipf(x)=0 for x =< a, = (x-a)/(b-a) for x between a and b, = 1 for x >= b
mathematical operation that corresponds to finding the area under the curve of a pdf f(x) from a to bthe integral of f(x) from a to b
The Normal DistributionThe normal distribution is completely described by two parameters—its mean, μ, and variance, σ2normal 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 0.16 As a consequence of symmetry, the mean, median, and the mode are all equal for a normal random variable
A multivariate distributionthe probabilities for a group of related random variables
There are two steps in standardizing a random variable XZ = (X – μ)/σ