## Continuous Random Variables

First, this section talks about how to describe continuous distributions and compute related probabilities, including some basic facts about the normal distribution. Then, it covers how to compute probabilities related to any normal random variable and gives examples of using -score transformations. Finally, it defines tail probabilities and illustrates how to find them.

### Probability Computations for General Normal Random Variables

#### Exercises

### Basic

1. is a normally distributed random variable with mean and standard deviation . Find the probability indicated.

3. is a normally distributed random variable with mean and standard deviation . Find the probability indicated.

5. is a normally distributed random variable with mean and standard deviation . Find the probability indicated.

7. is a normally distributed random variable with mean and standard deviation . Use Figure 12.2 "Cumulative Normal Probability" to find the first probability listed. Find the second probability using the symmetry of the density curve. Sketch the density curve with relevant regions shaded to illustrate the computation.

9. is a normally distributed random variable with mean and standard deviation . The probability that takes a value in the union of intervals will be denoted or . Use Figure 12.2 "Cumulative Normal Probability" to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the density curve you need to use Figure 12.2 "Cumulative Normal Probability" only one time for each part.

### Applications

11. The amount of beverage in a can labeled ounces is normally distributed with mean ounces and standard deviation ounce. A can is selected at random.

b. Find the probability that the can contains between and ounces.

13. The
systolic blood pressure of adults in a region is normally
distributed with mean and standard
deviation . A person is considered
"prehypertensive" if his systolic blood pressure is between and
. Find the probability that the blood
pressure of a randomly selected person is prehypertensive.

15.
Heights of adult men are normally distributed with mean
inches and standard deviation inches. Juliet, who is
inches tall, wishes to date only men who are taller than she but within
inches of her height. Find the probability that the next man she
meets will have such a height.

17. A
regulation golf ball may not weigh more than ounces. The
weights of golf balls made by a particular process are normally
distributed with mean ounces and standard deviation
ounce. Find the probability that a golf ball made by this process will
meet the weight standard.

19. The
amount of non-mortgage debt per household for households in a
particular income bracket in one part of the country is normally
distributed with mean and standard deviation .
Find the probability that a randomly selected such household has
between and in non-mortgage debt.

21. The
distance from the seat back to the front of the knees of seated adult
males is normally distributed with mean inches and standard
deviation inches. The distance from the seat back to the back
of the next seat forward in all seats on aircraft flown by a budget
airline is inches. Find the proportion of adult men flying with
this airline whose knees will touch the back of the seat in front of
them.

23. The useful life of a particular make and type of automotive tire is normally distributed with mean , miles and standard deviation miles.

a. Find the probability that such a tire will have a useful life of between and miles.

b.
Hamlet buys four such tires. Assuming that their lifetimes are
independent, find the probability that all four will last between
and miles. (If so, the best tire will have no more
than miles left on it when the first tire fails.) Hint: There
is a binomial random variable here, whose value of comes from
part (a).

25. The lengths of time taken by students on an algebra proficiency exam (if not forced to stop before completing it) are normally distributed with mean minutes and standard deviation minutes.

a. Find the proportion of students who will finish the exam if a -minute time limit is set.

b.
Six students are taking the exam today. Find the probability that all
six will finish the exam within the -minute limit, assuming that
times taken by students are independent. Hint: There is a binomial
random variable here, whose value of comes from part (a).

27. A regulation hockey puck must weigh between and ounces. In an alternative manufacturing process the mean weight of pucks produced is ounce. The weights of pucks have a normal distribution whose standard deviation can be decreased by increasingly stringent (and expensive) controls on the manufacturing process. Find the maximum allowable standard deviation so that at most of all pucks will fail to meet the weight standard. (Hint: The distribution is symmetric and is centered at the middle of the interval of acceptable weights).