Chapter 5 – Probability: The Mathematics of Randomness
Section 5.1: (Discrete) Random Variables
Definition 1
: A
random variable (rv)
is a quantity that (prior to observation) can be thought of
as dependent on chance phenomena.
We usually use capital letters at the end of the alphabet to
represent rv’s.
(X,Y,Z,W, etc.)
Definition 2
: A
discrete random variable
is one that has isolated or separated possible values
(rather than a continuum of available outcomes).
o
Example: Let X = number of head in 10 flips of a coin.
Here, the possible values for X are 0,1,2,…,10.
Thus, X is discrete
Definition 3
: A
continuous random variable
is one that can be idealized as having an entire
(continuous) interval of numbers as its set of possible values.
o
Example: Let Y = weight of babies born at a given hospital
Definition 4
: To specify a
probability distribution
for a rv is to give its set of possible values
and consistently assign numbers between 0 and 1, called probabilities, as measures of the
likelihood that the various numerical values will occur.
The tool most often used to describe a discrete probability distribution is the probability mass
function (pmf).
Definition 5
: A
probability mass function (pmf)
for a discrete random variable X, having
possible values
x
x
1
2
,
,
, is a nonnegative function
f
(
x
), with
f
x
j
(
)
giving the probability that
X takes the value
x
j
.
So
f
(
x
)=P[X =
x
] where X is a random variable and
x
is a specific numeric value.
(read as “
f
(
x
) equals the probability that X equals
x
)
Example 5.1
: Let X=the number of goals scored by a hockey team in each of their first 9 games
Suppose a team has X=1, 1, 0, 5, 0, 2, 8, 4, 1 goals.
•
Find the function
f
(
x
).
1