Chapter 6 –
The Binomial Probability Distribution
Defn
:
A random variable
is a variable whose values are determined by chance.
We will denote a
random variable by a capital letter, such as X, and denote particular values of the variable by the
corresponding lower case letter, x.
Thus we read P(X = x) as:
“the probability that the random
variable X takes on the value x.”
Defn
:
A discrete random variable
is a r.v. that has either a finite number of possible values or a
countable number of possible values.
Example
:
Consider the random experiment of rolling two dice, a green one and a red one.
Let the
variable X be the sum of the numbers showing on the top faces.
X can have the possible values 2, 3, 4,
5, 6, …, 11, or 12.
When we roll the two dice once, we cannot predict with certainty which value of X
will occur.
Example
:
Consider the population of all families in the United States.
Let the random variable X be
the number of children in a randomly selected family.
Then X has possible values 0, 1, 2, …, up to
some maximum number (certainly less than 24!).
Defn
:
A continuous random variable
is a r.v. that has an uncountably infinite number of possible
values.
Example
:
A man is randomly selected from the population of all adult males in the U.S.
Let X =
man’s height.
Defn
:
The probability distribution
of a r.v. X provides the possible values of X and their
corresponding probabilities.
A probability distribution may be in the form of a table or a mathematical
formula.
Example
:
For our random experiment of rolling two dice, if we assume that the dice are fair, then each
possible outcome in the sample space has the same probability of occurring.
If we let the random
variable X be the sum of the numbers on the top faces, then there is exactly one way for X to take on
the value 2, namely if the numbers showing on the top faces are both 1.
Thus
P(X = 2) = 1/36.
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 Fall '10
 Staff
 Statistics, Binomial, Probability, Probability distribution, Probability theory, Discrete probability distribution, Defn

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