## 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.

### Areas of Tails of Distributions

#### Exercises

### Basic

1. Find the value of that yields the probability shown.

a.b.

c.

d.

3. Find the value of that yields the probability shown.

5. Find the indicated value of . (It is easier to find and negate it.)

7. Find the value of that yields the probability shown, where is a normally distributed random variable with mean and standard deviation .

9. is a normally
distributed random variable with mean and standard
deviation . Find the values and of
that are symmetrically located with respect to the mean of and
satisfy . (Hint. First
solve the corresponding problem for .)

### Applications

11. Scores on a national exam are normally distributed with mean and standard deviation .

a. Find the score that is the th percentile.b. Find the score that is the th percentile.

13. The monthly amount of water used per household in a small community is normally distributed with mean gallons and standard deviation gallons. Find the three quartiles for the amount of water used.

15. Scores on the common final
exam given in a large enrollment multiple section course were normally
distributed with mean and standard deviation . The
department has the rule that in order to receive an A in the course his
score must be in the top of all exam scores. Find the minimum
exam score that meets this requirement.

17. Tests of a new tire developed
by a tire manufacturer led to an estimated mean tread life of
miles and standard deviation of miles. The
manufacturer will advertise the lifetime of the tire (for example, a "
mile tire") using the largest value for which it is expected
that of the tires will last at least that long. Assuming tire
life is normally distributed, find that advertised value.

19. The weights of eggs
produced at a particular farm are normally distributed with mean
ounces and standard deviation ounce. Eggs whose
weights lie in the middle of the distribution of weights of
all eggs are classified as "medium". Find the maximum and minimum
weights of such eggs. (These weights are endpoints of an interval that
is symmetric about the mean and in which the weights of of the
eggs produced at this farm lie.)

21. All students in a large
enrollment multiple section course take common in-class exams and a
common final, and submit common homework assignments. Course grades are
assigned based on students' final overall scores, which are
approximately normally distributed. The department assigns a to
students whose scores constitute the middle of all scores. If
scores this semester had mean and standard deviation ,
find the interval of scores that will be assigned a .

### Additional Exercises

23. A machine for filling -liter bottles of soft drink delivers an amount to each bottle that varies from bottle to bottle according to a normal distribution with standard deviation liter and mean whatever amount the machine is set to deliver.

a. If the machine is set to deliver liters (so the mean amount delivered is liters) what proportion of the bottles will contain at least liters of soft drink?b. Find the minimum setting of the mean amount delivered by the machine so that at least of all bottles will contain at least liters.