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CalculusCalculus: An Applied Approach (MindTap Course List)Finding Expected Value, Variance, and Standard Deviation In Exercises 1-12, use the given probability density function over the indicated interval to find the (a) expected value, (b) variance using the alternative formula, and (c) standard deviation of the random variable. (d) Then sketch the graph of the probability density function and locate the mean on the graph. See Examples 1, 2, and 5. f ( x ) = 6 x ( 1 − x ) , [ 0 , 1 ]BuyFind*launch*

10th Edition

Ron Larson

Publisher: Cengage Learning

ISBN: 9781305860919

Chapter 9.3, Problem 9E

Textbook Problem

**Finding Expected Value, Variance, and Standard Deviation** In Exercises 1-12, use the given probability density function over the indicated interval to find the (a) expected value,

(b) variance using the alternative formula, and

(c) standard deviation of the random variable.

(d) Then sketch the graph of the probability density function and locate the mean on the graph. See *Examples 1, 2, and 5.*

Calculus: An Applied Approach (MindTap Course List)

Ch. 9.1 - Checkpoint 1 Worked-out solution available at...Ch. 9.1 - Two coins are tossed. A random variable assigns...Ch. 9.1 - Checkpoint 3 Worked-out solution available at...Ch. 9.1 - Checkpoint 4 Worked-out solution available at...Ch. 9.1 - Over a period of one year, a salesperson worked...Ch. 9.1 - Find the variance and standard deviation of the...Ch. 9.1 - In Exercises 1 and 2, solve for x. 19+23+29=xCh. 9.1 - In Exercises 1 and 2, solve for x....Ch. 9.1 - In Exercises 3-6, evaluate the expression....Ch. 9.1 - In Exercises 3-6, evaluate the expression....

Ch. 9.1 - In Exercises 3-6, evaluate the expression....Ch. 9.1 - In Exercises 3-6, evaluate the expression....Ch. 9.1 - In Exercises 7-10, write the fraction as a...Ch. 9.1 - In Exercises 710, write the fraction as a percent....Ch. 9.1 - In Exercises 710, write the fraction as a percent....Ch. 9.1 - In Exercises 710, write the fraction as a percent....Ch. 9.1 - Finding Sample Spaces and events In Exercises 14,...Ch. 9.1 - Finding Sample Spaces and events In Exercises 14,...Ch. 9.1 - Finding Sample Spaces and events In Exercises 14,...Ch. 9.1 - Finding Sample Spaces and events In Exercises 14,...Ch. 9.1 - Finding Frequency and Probability Distributions In...Ch. 9.1 - Finding Frequency and Probability Distributions In...Ch. 9.1 - Random Selection In a class of 72 students, 44 are...Ch. 9.1 - Random Selection A card is chosen at random from a...Ch. 9.1 - Determining a Missing Probability In Exercises 9...Ch. 9.1 - Determining a Missing Probability In Exercises 9...Ch. 9.1 - Identifying Probability Distributions In Exercises...Ch. 9.1 - Identifying Probability Distributions In Exercises...Ch. 9.1 - Identifying Probability Distributions In Exercises...Ch. 9.1 - Identifying Probability Distributions In Exercises...Ch. 9.1 - Using Probability Distributions In Exercises...Ch. 9.1 - Using Probability Distributions In Exercises 1518,...Ch. 9.1 - Using Probability Distributions In Exercises...Ch. 9.1 - Using Probability Distributions In Exercises...Ch. 9.1 - Health The table shows the probability...Ch. 9.1 - Children The table shows the probability...Ch. 9.1 - Biology Consider a couple who have four children....Ch. 9.1 - Die Roll Consider the experiment of rolling a...Ch. 9.1 - Finding Expected Value, Variance, and Standard...Ch. 9.1 - Finding Expected Value, Variance, and Standard...Ch. 9.1 - Finding Expected Value, Variance, and Standard...Ch. 9.1 - Finding Expected Value, Variance, and Standard...Ch. 9.1 - Finding Mean, Variance, and Standard Deviation In...Ch. 9.1 - Finding Mean, Variance, and Standard Deviation In...Ch. 9.1 - Revenue A publishing company introduces a new...Ch. 9.1 - Personal Income The probability distribution of...Ch. 9.1 - Insurance An insurance company needs to determine...Ch. 9.1 - Insurance An insurance company needs to determine...Ch. 9.1 - Baseball A baseball fan examined the record of a...Ch. 9.1 - Games of Chance If x is a players net gain in a...Ch. 9.1 - Games of Chance If x is a players net gain in a...Ch. 9.1 - Market Analysis A sporting goods company has...Ch. 9.1 - Market Analysis Repeat Exercise 37 for the case in...Ch. 9.2 - Show that f(x)=12x is a probability density...Ch. 9.2 - Show that f(x)=2e2x is a probability density...Ch. 9.2 - Find the probability that x lies in the interval...Ch. 9.2 - A probability density function is defined over the...Ch. 9.2 - For the function in Example 5. find the...Ch. 9.2 - In Exercises 1-4, determine whether the function...Ch. 9.2 - In Exercises 1-4, determine whether the function...Ch. 9.2 - In Exercises 1-4, determine whether the function...Ch. 9.2 - In Exercises 1-4, determine whether the function...Ch. 9.2 - In Exercises 5-7, evaluate the definite or...Ch. 9.2 - In Exercises 5-7, evaluate the definite or...Ch. 9.2 - In Exercises 5-7, evaluate the definite or...Ch. 9.2 - Verifying a Probability Density Function In...Ch. 9.2 - Verifying a Probability Density Function In...Ch. 9.2 - Verifying a Probability Density Function In...Ch. 9.2 - Verifying a Probability Density Function In...Ch. 9.2 - Verifying a Probability Density Function In...Ch. 9.2 - Verifying a Probability Density Function In...Ch. 9.2 - Identifying Probability Density Functions In...Ch. 9.2 - Identifying Probability Density Functions In...Ch. 9.2 - Identifying Probability Density Functions In...Ch. 9.2 - Identifying Probability Density Functions In...Ch. 9.2 - Identifying Probability Density Functions In...Ch. 9.2 - Identifying Probability Density Functions In...Ch. 9.2 - Making a Probability Density Function In Exercises...Ch. 9.2 - Making a Probability Density Function In Exercises...Ch. 9.2 - Making a Probability Density Function In Exercises...Ch. 9.2 - Making a Probability Density Function In Exercises...Ch. 9.2 - Making a Probability Density Function In Exercises...Ch. 9.2 - Making a Probability Density Function In Exercises...Ch. 9.2 - Finding a Probability In Exercises 19-26, sketch...Ch. 9.2 - Finding a Probability In Exercises 19-26, sketch...Ch. 9.2 - Finding a Probability In Exercises 19-26, sketch...Ch. 9.2 - Finding a Probability In Exercises 19-26, sketch...Ch. 9.2 - Finding a Probability In Exercises 19-26, sketch...Ch. 9.2 - Finding a Probability In Exercises 19-26, sketch...Ch. 9.2 - Finding a Probability In Exercises 19-26, sketch...Ch. 9.2 - Finding a Probability In Exercises 19-26, sketch...Ch. 9.2 - Finding a Probability A probability density...Ch. 9.2 - Finding a Probability A probability density...Ch. 9.2 - Waiting Time Buses arrive and depart from a...Ch. 9.2 - Demand The daily demand for gasoline x (in...Ch. 9.2 - Learning Theory The time t (in hours) required for...Ch. 9.2 - HOW DO YOU SEE IT? The graph shows two models for...Ch. 9.2 - Using the Exponential Density Function In...Ch. 9.2 - Using the Exponential Density Function In...Ch. 9.2 - Using the Exponential Density Function In...Ch. 9.2 - Using the Exponential Density Function In...Ch. 9.2 - Meteorology A meteorologist predicts that the...Ch. 9.2 - Demand The weekly demand x (in tons) for a certain...Ch. 9.2 - Demand Using the probability density function in...Ch. 9.3 - Find the expected value of the probability density...Ch. 9.3 - Find the variance and standard deviation of the...Ch. 9.3 - Use a symbolic integration utility to find the...Ch. 9.3 - Find the mean and median of the probability...Ch. 9.3 - Find the expected value and standard deviation of...Ch. 9.3 - Find the probability that a person selected at...Ch. 9.3 - Checkpoint 7 Worked-out solution available at...Ch. 9.3 - In Exercises 1 and 2, use integration by parts to...Ch. 9.3 - In Exercises 1 and 2, use integration by parts to...Ch. 9.3 - In Exercises 3 and 4, evaluate the definite...Ch. 9.3 - In Exercises 3 and 4, evaluate the definite...Ch. 9.3 - In Exercises 5 and 6, find each probability using...Ch. 9.3 - In Exercises 5 and 6, find each probability using...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Finding Expected Value, Variance, and Standard...Ch. 9.3 - Using Two Methods In Exercises 13-16, find the...Ch. 9.3 - Using Two Methods In Exercises 13-16, find the...Ch. 9.3 - Using Two Methods In Exercises 13-16, find the...Ch. 9.3 - Using two Methods In Exercises 1316, find the...Ch. 9.3 - Using Technology In Exercises 17-22, use a...Ch. 9.3 - Using Technology In Exercises 17-22, use a...Ch. 9.3 - Using Technology In Exercises 17-22, use a...Ch. 9.3 - Using Technology In Exercises 17-22, use a...Ch. 9.3 - Using Technology In Exercises 17-22, use a...Ch. 9.3 - Using Technology In Exercises 17-22, use a...Ch. 9.3 - Finding the Mean and Median In Exercises 23-28,...Ch. 9.3 - Finding the Mean and Median In Exercises 23-28,...Ch. 9.3 - Finding the Mean and Median In Exercises 23-28,...Ch. 9.3 - Finding the Mean and Median In Exercises 23-28,...Ch. 9.3 - Finding the Mean and Median In Exercises 23-28,...Ch. 9.3 - Finding the Mean and Median In Exercises 23-28,...Ch. 9.3 - Special Probability Density Functions In Exercises...Ch. 9.3 - Special Probability Density Functions In Exercises...Ch. 9.3 - Special Probability Density Functions In Exercises...Ch. 9.3 - Special Probability Density Functions In Exercises...Ch. 9.3 - Special Probability Density Functions In Exercises...Ch. 9.3 - Special Probability Density Functions In Exercises...Ch. 9.3 - Finding Probabilities In Exercises 35 and 36, let...Ch. 9.3 - Finding Probabilities In Exercises 35 and 36, let...Ch. 9.3 - Consumer Trends The number of coupons x used by a...Ch. 9.3 - Cost The daily cost x (in dollars) of electricity...Ch. 9.3 - Useful Life The useful lifetime of a product is...Ch. 9.3 - Useful Life The useful lifetime of a product is...Ch. 9.3 - Transportation The arrival time t (in minutes) of...Ch. 9.3 - Transportation Repeat Exercise 41 for a bus that...Ch. 9.3 - Useful Life The time t (in years) until failure of...Ch. 9.3 - Useful Life The time t (in years) until failure of...Ch. 9.3 - Waiting Time The waiting time t (in minutes) for...Ch. 9.3 - License Renewal The waiting time t (in minutes) at...Ch. 9.3 - Demand The daily demand x for a certain product...Ch. 9.3 - Demand Repeat Exercise 47 for the probability...Ch. 9.3 - Demand The daily demand x for water (in millions...Ch. 9.3 - HOW DO YOU SEE IT? In chemistry, the probability...Ch. 9.3 - Wages The employees of a large corporation are...Ch. 9.3 - Medical Science A medical research team has...Ch. 9.3 - Education For high school graduates from 2012...Ch. 9.3 - Fuel Mileage Assume the fuel mileages of all 2015...Ch. 9 - Finding Sample Spaces and Events In Exercises 14,...Ch. 9 - Finding Sample Spaces and Events In Exercises 14,...Ch. 9 - Finding Sample Spaces and Events In Exercises 14,...Ch. 9 - Finding Sample Spaces and Events In Exercises 14,...Ch. 9 - Frequency Distributions In Exercises 5 and 6,...Ch. 9 - Frequency Distributions In Exercises 5 and 6,...Ch. 9 - Bar Code Use the frequency distribution in...Ch. 9 - Kittens Use the frequency distribution in Exercise...Ch. 9 - Identifying Probability Distributions In Exercises...Ch. 9 - Identifying Probability Distributions In Exercises...Ch. 9 - Using Probability Distributions In Exercises 11...Ch. 9 - Using Probability Distributions In Exercises 11...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Revenue A publishing company introduces a new...Ch. 9 - Raffle A service organization is selling $5 raffle...Ch. 9 - Consumer Trends The probability distribution of...Ch. 9 - Vital Statistics The probability distribution of...Ch. 9 - Verifying a Probability Density Function In...Ch. 9 - Verifying a Probability Density Function In...Ch. 9 - Verifying a Probability Density Function In...Ch. 9 - Verifying a Probability Density Function In...Ch. 9 - Identifying Probability Density Functions In...Ch. 9 - Identifying Probability Density Functions In...Ch. 9 - Identifying Probability Density Functions In...Ch. 9 - Identifying Probability Density Functions In...Ch. 9 - Identifying Probability Density Functions In...Ch. 9 - Identifying Probability Density Functions In...Ch. 9 - Finding a Probability In Exercises 31-36, sketch...Ch. 9 - Finding a Probability In Exercises 31-36, sketch...Ch. 9 - Finding a Probability In Exercises 31-36, sketch...Ch. 9 - Finding a Probability In Exercises 31-36, sketch...Ch. 9 - Finding a Probability In Exercises 31-36, sketch...Ch. 9 - Finding a Probability In Exercises 31-36, sketch...Ch. 9 - Waiting Time The waiting time t (in minutes) for...Ch. 9 - Medicine The time t (in days) until recovery after...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding Expected Value, Variance, and Standard...Ch. 9 - Finding the Mean and Median In Exercises 4548,...Ch. 9 - Finding the Mean and Median In Exercises 4548,...Ch. 9 - Finding the Mean and Median In Exercises 4548,...Ch. 9 - Finding the Mean and Median In Exercises 4548,...Ch. 9 - Special Probability Density Functions In Exercises...Ch. 9 - Special Probability Density Functions In Exercises...Ch. 9 - Special Probability Density Functions In Exercises...Ch. 9 - Special Probability Density Functions In Exercises...Ch. 9 - Special Probability Density Functions In Exercises...Ch. 9 - Special Probability Density Functions In Exercises...Ch. 9 - Transportation The arrival time t (in minutes) of...Ch. 9 - Transportation Repeat Exercise 55 for a train that...Ch. 9 - Waiting Time The waiting time t (in minutes) for...Ch. 9 - Useful Life The time t (in hours) until failure of...Ch. 9 - Demand The daily demand x for a certain product...Ch. 9 - Demand Repeat Exercise 59 for the probability...Ch. 9 - Education In 2013, the scores on the Medical...Ch. 9 - Botany In a botany experiment, plants are grown in...Ch. 9 - Four students answer a true-false question on an...Ch. 9 - A card is chosen at random from a standard 52-card...Ch. 9 - In Exercises 3 and 4, sketch a graph of the...Ch. 9 - In Exercises 3 and 4, sketch a graph of the...Ch. 9 - In Exercises 5 and 6, find the expected value,...Ch. 9 - In Exercises 5 and 6, find the expected value,...Ch. 9 - In Exercises 79, use a graphing utility to graph...Ch. 9 - In Exercises 79, use a graphing utility to graph...Ch. 9 - In Exercises 79, use a graphing utility to graph...Ch. 9 - In Exercises 1012, use the given probability...Ch. 9 - In Exercises 1012, use the given probability...Ch. 9 - In Exercises 1012, use the given probability...Ch. 9 - In Exercises 1315, use the given probability...Ch. 9 - In Exercises 1315, use the given probability...Ch. 9 - In Exercises 1315, use the given probability...Ch. 9 - An intelligence quotient or IQ is a number that is...

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