The ** probability distribution** of a discrete random variable
is a list of probabilities associated with each of its possible values.
It is also sometimes called the probability function or the probability
mass function.

(*Definitions taken from Valerie J. Easton and John H. McColl's
Statistics Glossary v1.1*)

Suppose a random variable *X* may take *k* different
values, with the probability that *X = x _{i}* defined to be

**1:**0__<__p_{i}__<__1 for each i**2:**p_{1}+ p_{2}+ ... + p_{k}= 1.

The probabilities associated with each outcome are described by the following table:

Outcome 1 2 3 4 Probability 0.1 0.3 0.4 0.2The probability that

This distribution may also be described by the ** probability
histogram** shown to the right:

All random variables (discrete and continuous) have a

(*Definition taken from Valerie J. Easton and John H. McColl's
Statistics Glossary v1.1*)

The probability that X is less than or equal to 1 is 0.1,

the probability that X is less than or equal to 2 is 0.1+0.3 = 0.4,

the probability that X is less than or equal to 3 is 0.1+0.3+0.4 = 0.8, and

the probability that X is less than or equal to 4 is 0.1+0.3+0.4+0.2 = 1.

The probability histogram for the cumulative distribution of this
random variable is shown to the right:

(*Definition taken from Valerie J. Easton and John H. McColl's
Statistics Glossary v1.1*)

A continuous random variable is not defined at specific values. Instead,
it is defined over an *interval* of values, and is represented by
the * area under a curve* (in advanced mathematics, this is
known as an

Suppose a random variable *X* may take all values over an interval of
real numbers. Then the probability that *X* is in the
set of outcomes *A, P(A)*, is defined to be the area above *A* and
under a curve. The curve, which represents a function *p(x)*, must
satisfy the following:

**1:**The curve has no negative values (p(x)__>__0 for all x)**2:**The total area under the curve is equal to 1.

A curve meeting these requirements is known as a * density curve*.

The following graphs plot the density curves for random number generators
over the intervals (4,5) (top left), (2,6) (top right), (5,5.5) (lower left),
and (3,5) (lower right). The distributions corresponding to these curves are
known as * uniform distributions*.

The uniform distribution is often used to simulate data. Suppose you would like to simulate data for 10 rolls of a regular 6-sided die. Using the MINITAB "RAND" command with the "UNIF" subcommand generates 10 numbers in the interval (0,6):

MTB > RAND 10 c2; SUBC> unif 0 6.Assign the discrete random variable X to the values 1, 2, 3, 4, 5, or 6 as follows:

if 0<X<1, X=1

if 1<X<2, X=2

if 2<X<3, X=3

if 3<X<4, X=4

if 4<X<5, X=5

if X>5, X=6.

Use the generated MINITAB data to assign X to a value for each roll of the die:

Uniform Data X Value 4.53786 5 5.77474 6 3.69518 4 1.03929 2 4.23835 5 0.37096 1 0.75272 1 5.56563 6 0.89045 1 3.18086 4

Another type of continuous density curve is the normal distribution. The area under the curve is not easy to calculate for a normal random variable