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 \(z\)-score transformations. Finally, it defines tail probabilities and illustrates how to find them.
Continuous Random Variables
Exercises
Basic
1. A continuous random variable \(X\) has a uniform distribution on the interval \([5,12]\). Sketch the graph of its density function.
3. A continuous random variable \(X\) has a normal distribution with mean \(100\) and standard deviation \(10\). Sketch a qualitatively accurate graph of its density function.
5. A continuous random variable \(X\) has a normal distribution with mean \(73\). The probability that \(X\) takes a value greater than \(80\) is \(0.212\). Use this information and the symmetry of the density function to find the probability that \(X\) takes a value less than \(66\). Sketch the density curve with relevant regions shaded to illustrate the computation.
7. A continuous random variable \(X\) has a normal distribution with mean \(50.5\). The probability that \(X\) takes a value less than \(54\) is \(0.76\). Use this information and the symmetry of the density function to find the probability that \(X\) takes a value greater than \(47\). 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 \(X_{A}\), \(X_{B}\), and \(X_{C}\). Their standard deviations (in no particular order) are \(15\), \(7\), and \(20\). Use the figure to identify the values of the means \(\mu_{A}\), \(\mu_{B}\), and \(\mu_{C}\) and standard deviations \(\sigma_{A}\), \(\sigma_{B}\), and \(\sigma_{C}\) 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 \(X\) of orange juice in a randomly selected half-gallon container varies according to a normal distribution with mean \(64\) ounces and standard deviation \(0.25\) ounce.
a. Sketch the graph of the density function for \(X\).b. What proportion of all containers contain less than a half gallon (\(64\) ounces)? Explain.
c. What is the median amount of orange juice in such containers? Explain.