A random variable assigns a numerical value to each outcome of a random experiment. It can be discrete or continuous depending on the values it can take. A discrete random variable takes on countable values like the number of heads in a coin toss, while a continuous variable can take any value in an interval like height. The probability mass function (pmf) of a discrete random variable gives the probability of each possible value and must be non-negative and sum to 1. Examples show how to construct pmfs for different experiments and verify they satisfy the conditions.

1. * A random variable is a rule that assigns one value to
each point in a sample space for an experiment.
* A random variable can be classified as discrete or
continuous depending on the numerical values
it assumes.
1. A discrete random variable may assume either finite
or infinite sequence of values.
2. A continuous random variable may assume any
numerical in an interval or collection of interval.

2. 1. number of children in a family
2. Friday night attendance at a cinema
3. number of patients in a doctor's surgery
4. number of defective light bulbs in a box of ten.
1. height of students in class
2. weight of students in class
3. time it takes to get to school
4. distance traveled between classes

3. Consider an experiment of tossing a coin three times.
S = {HHH, HHT, HTT, HTH, THT, THH, TTH, TTT}
Let X assign to each sample point on S the total
number of head occurs. Then X is random variable
with range space Rx = { 0, 1, 2, 3}, since range space is
finite, X, is a discrete random variable.

4. union
GIRL
BOY
GIRL
BOY
BOY
GIRL
Couple plans to have 2 children. The random
circumstance includes the 2 births, specifically the sexes of
the 2 children. Let X assign to each sample point on S the
number of girls.
S = {BB, BG, GB, GG}
X is a random variable
with range Rx = {0, 1, 2},
since Rx is finite X is a
discrete random variable.

5. If X is a random variable, the function given by
f(x) = P [ X = x ] for each X within the range of X is called
the probability mass function (pmf ) of X.
A function can serve as probability mass function of a
discrete random variable X if and only if its values f(x),
satisfies the following conditions:
1. f(x)  0 for all values of x;
2.  of all f(x) is equal to 1.

6. To express the probability mass function, we will
construct a table that exhibits the correspondence between
the values of random variables and the associated
probabilities.
Consider example # 1. The experiment consisting of
three tosses of a coin, assume that all 8 outcomes are
equally likely then the probability mass function for the
total number of heads is:
X 0 1 2 3
f(X) 1/8 3/8 3/8 1/8

7. To show that is a probability mass
function (pmf):
Condition 1:
Notice that all f(x) are all greater than
or equal to zero ;
Condition 2:
The sum of all f(x) is 1, that is:
1/8 + 3/8 + 3/8 + 1/8 = 1.

8. Referring to example # 2.
The experiment on the plan of the couple who wanted
to have two children. There are 6 possible outcomes
belonging to the sample space S. Let X assign to each sample
point on S the number of girls.
X is a random variable defined by a function f(x) = P [
X =x ], thus it would be a (pmf) such that:
x 0 1 2
f(x) ¼ ½ ¼

9. Condition 1 is satisfied:
for all f(x) it is greater than or equal to zero;
Condition 2 is satisfied:
the sum of all f(x) is equal to 1, that is
 (f (x)) = ¼ + ½ + ¼
= 1.