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Textbook by: Kathy Chu. E-mail the author

# Exercises

Module by: Kathy Chu. E-mail the author

Summary: This module provides a number of homework exercises related to Continuous Random Variables.

For each probability and percentile problem, DRAW THE PICTURE!

## Exercise 1

Consider the following experiment. You are one of 100 people enlisted to take part in a study to determine the percent of nurses in America with an R.N. (registered nurse) degree. You ask nurses if they have an R.N. degree. The nurses answer “yes” or “no.” You then calculate the percentage of nurses with an R.N. degree. You give that percentage to your supervisor.

• a. What part of the experiment will yield discrete data?
• b. What part of the experiment will yield continuous data?

## Exercise 2

When age is rounded to the nearest year, do the data stay continuous, or do they become discrete? Why?

## Exercise 3

Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a Uniform Distribution from 1 – 53 (spread of 52 weeks).

• a. X X size 12{X} {} ~
• b. Graph the probability distribution.
• c. f ( x ) f ( x ) size 12{f $$x$$ } {} =
• d. μ μ size 12{μ} {} =
• e. σ σ size 12{σ} {} =
• f. Find the probability that a person is born at the exact moment week 19 starts. That is, find P ( X = 19 ) P ( X = 19 ) size 12{P $$X="19"$$ } {} .
• g. P ( 2 < X < 31 ) = P ( 2 < X < 31 ) = size 12{P $$2<X<"31"$$ ={}} {}
• h. Find the probability that a person is born after week 40.
• i. {} P ( 12 < X X < 28 ) P ( 12 < X X < 28 ) size 12{P $$"12"<X \lline X<"28"$$ } {} =
• j. Find the 70th percentile.
• k. Find the minimum for the upper quarter.

### Solution

• a. X ~ U ( 1, 53 ) X ~ U ( 1, 53 ) size 12{X " ~ " U $$1,"53"$$ } {}
• c. f(x)=152f(x)=152 size 12{f $$x$$ = { {1} over { $$b - a$$ } } = { {1} over { $$"53" - 1$$ } } = { {1} over {"52"} } } {} where 1x531x53 size 12{1 <= x <= "53"} {}
• d. 27
• e. 15.01
• f. 0
• g. 29522952
• h. 13521352
• i. 16271627
• j. 37.4
• k. 40

## Exercise 4

A random number generator picks a number from 1 to 9 in a uniform manner.

• a. X ~ X ~ size 12{X "~" } {}
• b. Graph the probability distribution.
• c. f ( x ) = f ( x ) = size 12{f $$x$$ ={}} {}
• d. μ = μ = size 12{μ={}} {}
• e. σ = σ = size 12{σ={}} {}
• f. P ( 3 . 5 < X < 7 . 25 ) = P ( 3 . 5 < X < 7 . 25 ) = size 12{P $$3 "." 5<X<7 "." "25"$$ ={}} {}
• g. P ( X > 5 . 67 ) = P ( X > 5 . 67 ) = size 12{P $$X>5 "." "67"$$ ={}} {}
• h. P ( X > 5 X > 3 ) = P ( X > 5 X > 3 ) = size 12{P $$X>5 \lline X>3$$ ={}} {}
• i. Find the 90th percentile.

## Exercise 5

The speed of cars passing through the intersection of Blossom Hill Road and the Almaden Expressway varies from 10 to 35 mph and is uniformly distributed. None of the cars travel over 35 mph through the intersection.

• a. X = X = size 12{X={}} {}
• b. X ~ X ~ size 12{X "~" } {}
• c. Graph the probability distribution.
• d. f ( x ) = f ( x ) = size 12{f $$x$$ ={}} {}
• e. μ = μ = size 12{μ={}} {}
• f. σ = σ = size 12{σ={}} {}
• g. What is the probability that the speed of a car is at most 30 mph?
• h. What is the probability that the speed of a car is between 16 and 22 mph.
• i. P(20<X<53)=P(20<X<53)= size 12{P $$"20"<X<"53"$$ ={}} {} State this in a probability question (similar to g and h ), draw the picture, and find the probability.
• j. Find the 90th percentile. This means that 90% of the time, the speed is less than _____ mph while passing through the intersection per minute.
• k. Find the 75th percentile. In a complete sentence, state what this means. (See j.)
• l. Find the probability that the speed is more than 24 mph given (or knowing that) it is at least 15 mph.

### Solution

• b. X ~ U ( 10 , 35 ) X ~ U ( 10 , 35 ) size 12{X "~" U $$"10","35"$$ } {}
• d. f(x)=125f(x)=125 where 10X3510X35
• e. 452452
• f. 7.22
• g. 4545
• h. 625625
• i. 3535
• j. 32.5
• k. 28.75
• l. 11201120

## Exercise 6

According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between 6 and 15 pounds a month until they approach trim body weight. Let’s suppose that the weight loss is uniformly distributed. We are interested in the weight loss of a randomly selected individual following the program for one month. (Source: The McDougall Program for Maximum Weight Loss by John A. McDougall, M.D.)

• a. X = X = size 12{X={}} {}
• b. X ~ X ~ size 12{X "~" } {}
• c. Graph the probability distribution.
• d. f ( x ) = f ( x ) = size 12{f $$x$$ ={}} {}
• e. μ = μ = size 12{μ={}} {}
• f. σ = σ = size 12{σ={}} {}
• g. Find the probability that the individual lost more than 10 pounds in a month.
• h. Suppose it is known that the individual lost more than 10 pounds in a month. Find the probability that he lost less than 12 pounds in the month.
• i. P ( 7 < X < 13 X > 9 ) = P ( 7 < X < 13 X > 9 ) = size 12{P $$7<X<"13" \lline X>9$$ ={}} {} State this in a probability question (similar to g and h), draw the picture, and find the probability.

## Exercise 7

A subway train on the Red Line arrives every 8 minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution.

• a. X = X = size 12{X={}} {}
• b. X ~ X ~ size 12{X "~" } {}
• c. Graph the probability distribution.
• d. f ( x ) = f ( x ) = size 12{f $$x$$ ={}} {}
• e. μ = μ = size 12{μ={}} {}
• f. σ = σ = size 12{σ={}} {}
• g. Find the probability that the commuter waits less than one minute.
• h. Find the probability that the commuter waits between three and four minutes.
• i. 60% of commuters wait more than how long for the train? State this in a probability question (similar to g and h), draw the picture, and find the probability.

### Solution

• b. X ~ U ( 0,8 ) X ~ U ( 0,8 ) size 12{X "~" U $$0,8$$ } {}
• d. f(x)=18f(x)=18 where 0X80X8
• e. 4
• f. 2.31
• g. 1818
• h. 1818
• i. 3.2

## Exercise 8

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. X = X = size 12{X={}} {}
• b. X ~ X ~ size 12{X "~" } {}
• c. Graph the probability distribution.
• d. f ( x ) = f ( x ) = size 12{f $$x$$ ={}} {}
• e. μ = μ = size 12{μ={}} {}
• f. σ = σ = size 12{σ={}} {}
• g. Find the probability that she is over 6.5 years.
• h. Find the probability that she is between 4 and 6 years.
• i. Find the 70th percentile for the age of first graders on September 1 at Garden Elementary School.

## Try these multiple choice problems

The next three questions refer to the following information. The Sky Train from the terminal to the rental car and long term parking center is supposed to arrive every 8 minutes. The waiting times for the train are known to follow a uniform distribution.

### Exercise 9

What is the average waiting time (in minutes)?

• A. 0.0000
• B. 2.0000
• C. 3.0000
• D. 4.0000

D

### Exercise 10

Find the 30th percentile for the waiting times (in minutes).

• A. 2.0000
• B. 2.4000
• C. 2.750
• D. 3.000

B

### Exercise 11

The probability of waiting more than 7 minutes given a person has waited more than 4 minutes is?

• A. 0.1250
• B. 0.2500
• C. 0.5000
• D. 0.7500

B

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