Random Variables and Discrete Probability Distribution

Discrete

The random variable assumes a countable number of distinct values.

expected value

mean

Continuous

The random variable is characterized by (infinitely) uncountable values within any interval.

property of discrete probability distributions

The probability of each value x is a value between 0 and 1, or equivalently 0 <= P(X= x) <= 1

property of discrete probability distributions

The sum of the probabilities equals 1. In other words, Σ P(X <= xi) = 1 where the sum extends over all values x of X.

Discrete Random Variable

assumes a countable number of distinct values

discrete uniform distribution characteristics

distribution has a finite number of specified values each value is equally likely distribution is symmetric

cumulative distribution function

for X is defined as P(X <= x)

probability mass function

for a discrete random variable X is a list of the values of X with the associated probabilities,

Random Variable

is a function that assigns numerical values to the outcomes of an experiment

Every random variable

is associated with a probability distribution that describes the variable completely.

E(X)

is the long-run average value of the random variable over infinitely many independent repetitions of an experiment.

probability density function

is used to describe continuous random variables.

probability mass function

is used to describe discrete random variables.

Values of Random Variable

lower case

discrete probability distribution

may be viewed as a table, algebraically, or graphically.

cumulative distribution function

maybe used to describe either discrete or continuous random variables.

probability mass function

the list of all possible pairs (x,P(X= x)

Random Variable

upper case

Values of Random Variable

x1, x2,x3...

E(X)

Σ xiP(X <= xi)