Chapter 5: Continuous Random Variables

In major league baseball, a no-hitter is a game in which a pitcher, or pitchers, doesn't give up any hits throughout the game. No-hitters occur at a rate of about three per season. Assume that the duration of time between no-hitters is exponential.

a. What is the probability that an entire season elapses with a single no-hitter?

b. If an entire season elapses without any no-hitters, what is the probability that there are no no-hitters in the following season?

c. What is the probability that there are more than $3$ no-hitters in a single season?

98. During the years 1998-2012, a total of 29 earthquakes of magnitude greater than 6.5 have occurred in Papua New Guinea. Assume that the time spent waiting between earthquakes is exponential.

a. What is the probability that the next earthquake occurs within the next three months?

b. Given that six months have passed without an earthquake in Papua New Guinea, what is the probability that the next three months will be free of earthquakes ?

c. What is the probability of zero earthquakes occurring in 2014?

d. What is the probability that at least two earthquakes will occur in 2014 ?

According to the American Red Cross, about one out of nine people in the U.S. have Type B blood. Suppose the blood types of people arriving at a blood drive are independent. In this case, the number of Type B blood types that arrive roughly follows the Poisson distribution.

a. If $100$ people arrive, how many on average would be expected to have Type B blood?

b. What is the probability that over $10$people out of these 100 have type B blood?

c. What is the probability that more than $20$ people arrive before a person with type B blood is found?

The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from $5.8$to $6.8$years.

We randomly select one first grader from the class.

a. Define the random variable. $X$= _________

b. $X~$_________

c. Graph the probability distribution.

d.$f\left(x\right)$ = _________

e.$\mu$= _________

f. $\sigma$= _________

g. Find the probability that she is over $6.5$years old.

h. Find the probability that she is between four and six years old.

i. Find the ${70}^{th}$percentile for the age of first graders on September 1 at Garden Elementary School.

Use the following information to answer the next three exercises.

The Sky Train from the terminal to the rental–car and long–term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution.

79. What is the average waiting time (in minutes)?

a. zero

b. two

c. three

d. four

Use the following information to answer the next three exercises.

The Sky Train from the terminal to the rental–car and long–term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution.

80. Find the ${30}^{th}$percentile for the waiting times (in minutes).

a. two

b.$2.4$

c.$2.75$

d. three

Use the following information to answer the next three exercises.

The Sky Train from the terminal to the rental–car and long–term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution.

The probability of waiting more than seven minutes given a person has waited more than four minutes is?

a. $0.125$

b. $0.25$

c. $0.5$

d. $0.75$

Suppose that the value of a stock varies each day from $\backslash (16\mathrm{to}\backslash )25$with a uniform distribution.

a. Find the probability that the value of the stock is more than $\backslash (19$.

b. Find the probability that the value of the stock is between role="math" localid="1648188993020" $\backslash )19\mathrm{and}\backslash (22$.

c. Find the upper quartile - $25\%$of all days the stock is above what value? Draw the graph.

d. Given that the stock is greater than $\backslash )18$, find the probability that the stock is more than $\$21$.

A fireworks show is designed so that the time between fireworks is between one and five seconds, and follows a uniform

distribution.

a. Find the average time between fireworks.

b. Find probability that the time between fireworks is greater than four seconds.

The number of miles driven by a truck driver falls between $300\mathrm{and}700$, and follows a uniform distribution.

a. Find the probability that the truck driver goes more than $650$miles in a day.

b. Find the probability that the truck drivers goes between $400\mathrm{and}650$miles in a day.

c. At least how many miles does the truck driver travel on the furthest $10\%$of days?