Chapter 5
Discrete Probability Distributions
5-2
Random Variables
1. As defined in the text, a random variable is a variable that takes on a single numerical value,
determined by chance, for each outcome of a procedure.
In this exercise, the random variable
is the number of winning lottery tickets obtained over a 52 week period.
The possible values
for this random variable are 0,1,2,…,52.
2. No; the expected value does not have to be one of the values that the random variable can
actually take on.
The expected value is not the value most likely to occur on any given trial,
but rather it is mean of the values obtained from an infinite number of trials (i.e., the
weighted
mean of the all the values in the population).
3. The probability distribution gives each possible non-overlapping outcome O
i
of a procedure
and the probability P(O
i
) associated with each of those outcomes.
Since it is a certainty that one of the outcomes will occur,
P(O
1
or O
2
or O
3
or…) = 1.
Since the outcomes are mutually exclusive,
P(O
1
or O
2
or O
3
or…) = P(O
1
) +P(O
2
) +P(O
3
) +….
Therefore,
P(O
1
) +P(O
2
) +P(O
3
) +… =
Σ
P(O
i
) = 1.
4. No; such a loading of a die is not possible because it would lead to probabilities larger than 1 –
e.g., P(x=5 or x=6) = P(x=5) + P(x=6) = 0.5 + 0.6 = 1.1.
No; the listing of these outcomes and
their supposed probabilities does not constitute a probability distribution because the sum of
the probabilities is 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 = 2.1, which is greater than 1.
5. a. Discrete, since such a number is limited to certain values – viz., the non-negative integers.
b. Continuous, since weight can be any value on a specified continuum.
c. Continuous, since height can be any value on a specified continuum.
d. Discrete, since such a number is limited to certain values – viz., the non-negative integers.
e. Continuous, since time can be any value on a specified continuum.
6. a. Continuous, since volume can be any value on a specified continuum.
b. Discrete, since such a number is limited to certain values – viz., the non-negative integers.
NOTE: One could make a case that a person could consume 1/2 of a can, or 3/4 of a can, or
any portion of a can [even 1/
2 = 0.7071… of a can] – so that the number of cans
consumed is a continuous random variable.
c. Discrete, since such a number is limited to certain values – viz., the non-negative integers.
d. Continuous, since time can be any value on a specified continuum.
e. Discrete, since such costs are limited to certain values – viz., whole numbers of cents.
NOTE: If one of the conditions for a probability distribution does not hold, the formulas do not
apply – and they produce numbers that have no meaning. When working with probability
distributions and formulas in the exercises that follow, always keep the following important facts
in mind.