Discrete Probability Distributions
Definition: A random variable is a variable whose values are determined by
chance.
Definition: When the observations of a quantitative random variable can take on
only a finite number of values or a countable number of values, we say that the
variable is a discrete random variable.
Definition: A discrete probability distribution consists of the values a random
variable can assume and the corresponding probabilities of the values.
The probability distribution is simply an assignment of probabilities to the specific
values of the random variable or to a range of values of the random variable. The
sum of the probabilities must be 1 since they correspond to every possible value.
EXAMPLE: The probabilities of a return on an investment of $1000, $2000, and
$3000 are ½, ¼, and ¼ respectively. Construct a probability distribution.
EXAMPLE: Three patients are given a headache relief tablet. The probabilities
for 0, 1, 2, or 3 successes are 0.18, 0.52, 0.21, and 0.09, respectively. Construct a
probability distribution.
EXAMPLE: Construct a probability distribution for a family of three children.
Let X represent the number of boys.
EXAMPLE: Construct a probability distribution for the number of heads after
tossing a coin three times.
HOMEWORK: p250, 6 through 29 odd
Mean Variance and Expectation
Formula for the Mean of a Probability Distribution
The mean of the random variable with a discrete probability distribution is
µ = X1 · P(X1) + X2 · P(X2) + X3 · P(X3) + … + Xn · P(Xn)
or
µ = ∑ X · P(X)
where X1, X2, X3, …, Xn are the outcomes and P(X1), P(X2), P(X3), …, P(Xn) are
the corresponding probabilities and  is the expected value of x.
EXAMPLE: Find the mean number of boys in a family of three children.
Number of Boys, X
0
1
2
3
Probability, P(X) 0.125 0.375 0.375 0.125
EXAMPLE: The probability distribution for the number of customers per day at a
coffee shop is shown. Find the mean.
Number of Customers, X 50
51
52
53
54
Probability, P(X)
0.10 0.20 0.37 0.21 0.12
Formulas for the Variance and Standard Deviation of a Probability Distribution
The standard deviation of a probability distribution is
  [ X 2  P( X )]   2
The formula for the variance of a probability distribution is
σ2 = ∑ [X2 · P(X)] – µ2
EXAMPLE: Find the variance and standard deviation for the number of boys in a
family of three children.
Number of Boys, X
0
1
2
3
Probability, P(X) 0.125 0.375 0.375 0.125
EXAMPLE: The probability distribution for the number of customers per day at a
coffee shop is shown. Find the variance and standard deviation.
Number of Customers, X 50
51
52
53
54
Probability, P(X)
0.10 0.20 0.37 0.21 0.12
Definition: The expected value of a discrete random variable of a probability
distribution is the theoretical average of the variable. The formula is
E(X) = µ = ∑ X · P(X)
EXAMPLE: A box contains ten $1 bills, five $2 bills, three $5 bills, one $10 bill,
and one $100 bill. A person is charged $20 to select one bill. Find the
expectation.
EXAMPLE: If a player rolls two dice and gets a sum of 2 or 12, she wins $20. If
the person gets a 7, she wins $5. The cost to play the game is $3. Find the
expected value of the game.
HOMEWORK: p 259, 1 through 17 odd