## 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.

### Continuous Random Variables

#### Exercises

### Basic

1. A continuous random variable has a uniform distribution on the interval . Sketch the graph of its density function.

3. A continuous random variable has a normal distribution with mean and standard deviation . Sketch a qualitatively accurate graph of its density function.

5. A continuous random variable has a normal distribution with mean . The probability that takes a value greater than is . Use this information and the symmetry of the density function to find the probability that takes a value less than . Sketch the density curve with relevant regions shaded to illustrate the computation.

7. A continuous random variable has a normal distribution with mean . The probability that takes a value less than is . Use this information and the symmetry of the density function to find the probability that takes a value greater than . Sketch the density curve with relevant regions shaded to illustrate the computation.

9. The figure provided shows the density curves of three normally distributed random variables , , and . Their standard deviations (in no particular order) are , , and . Use the figure to identify the values of the means , , and and standard deviations , , and of the three random variables.

### Applications

11. Dogberry's alarm clock is battery operated. The battery could fail with equal probability at any time of the day or night. Every day Dogberry sets his alarm for 6:30 a.m. and goes to bed at 10:00 p.m. Find the probability that when the clock battery finally dies, it will do so at the most inconvenient time, between 10:00 p.m. and 6:30 a.m.

13. The amount of orange juice in a randomly selected half-gallon container varies according to a normal distribution with mean ounces and standard deviation ounce.

a. Sketch the graph of the density function for .b. What proportion of all containers contain less than a half gallon ( ounces)? Explain.

c. What is the median amount of orange juice in such containers? Explain.