Elementary Statistics –
Random Variable (Distribution, Mean, and Variance) Examples
Example 1 (Discrete random variable)
A fair 3-sided die is
rolled twice, and each time the up-face is recorded. Let X be the sum of
the two rolls.
- Determine the probability distribution of X.
- Determine the probability of getting a sum of
4 or greater.
- Determine the mean (also called expected
value) of X (this is denoted
).
Law of Large
Numbers
Suppose independent
observations are drawn at random from any population. As the number of
observations increases, the sample mean
of the observed values gets closer and closer to the mean
of the population. (Hence, is the long-run average of many independent
observations of the variable.)
- Determine the variance and standard deviation
of X (denoted
and 
, respectively).
Example 2 (Continuous random variable)
Suppose the distribution
of scores on a standardized exam is approximately normal with mean 27 points
and standard deviation 5 points. Let X be the score of a randomly
selected exam. Then X has the N(27, 5) distribution. Find the
probability that X exceeds 30 points.
[Important notes: 1) The mean and standard deviation are provided
within the problem; we do not need to derive them. For continuous random
variables, the calculation of means and variances involves integration—a
technique from calculus. 2) The calculations involving the normal curve are
exactly the same as in Chapter 1. Now we simply use more mathematical notation
(areas under the curve represent probabilities) and consider the normal curve
as a model for a random variable, rather than for a set of data.]
Example 3
(Illustrating rule 1 for means)
An insurance company offers a combination home/auto policy that insures against both home fire damage and auto hail damage. The policy pays $20,000 for home fire damage and $1000 for auto hail damage (this is a very simple policy). According to the company’s records, the probability (per year) of home fire damage is 0.01 and the probability (per year) of auto hail damage is 0.05. Furthermore, the events of fire damage and hail damage are independent, and the company covers a maximum of 1 home fire per year and 1 auto hail damage per year.
How much should be charged for the combination home/auto insurance policy so the company has an expected net profit of $5 per policy?
Example 4
(Illustrating the combined use of rules
1 and 2 for means and variances)
Suppose the distribution
of Jane’s possible scores on a standardized science exam has mean 75 points and
standard deviation 9 points (there is variability in her scores, because of how
she’s feeling, what she’s studied recently, etc.). Suppose the distribution of
John’s possible scores on the exam has mean 84 points and standard deviation 5
points. Furthermore, suppose Jane and John’s scores are independent. To create
their team score, Jane’s score is doubled and John’s score is tripled, and then
the two new values are added together (i.e.,
2X + 3Y, where X is Jane’s score and Y is John’s score).
Find the mean and standard deviation of their team score.