Discrete Random Variables & Distributions

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Discrete Random Variables & Distributions

A random variable X is a function (or rule) that assigns to each possible outcome s in the sample space S, a real number X(s). It is a numerical outcome from a random process.

Discrete if X takes values in some discrete set (countable).
Continuous if X can take on any value in some given range

The Probability Distribution for a random variable X tells us how the total probability of 1 for the sample space S is distributed among each of the mutually exclusive simple events (or outcomes for X) that describe the sample space. The distribution consists of a list of all possible values for X along with the probability that X assumes each of these. The function that describes these probabilities is called the probability mass function (or the p.m.f) and denoted by p(x), where

p(x) = P(X=x) = P(all sÎ S| X(s)=x)

Usually, the distribution is visually displayed by a population histogram (also called a probability histogram).

A parameter in a probability distribution is some constant that appears in p(x), (the p.m.f). Different values for the parameter could lead to different functions p(x) and the set of all functions that may be obtained by varying the value of the parameter is called a family of probability distributions. (Note: Some distributions have more than one parameter.)

The cumulative distribution function (or cdf) F(x) for a random variable X with p.m.f. p(x) is the probability that X is no larger than x and is defined as

F(x) = P(X£ x) = å_y|y£
x p(y)

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Jayant Rajgopal