Calculus: An Applied Approach (Min...

10th Edition

Ron Larson

ISBN: 9781305860919

Textbook Problem

1 views

Verifying a Probability Density Function In Exercises 1-6, show that the function is a probability density function over the given interval. See Examples 1 and 2.

f ( x ) = 4 27 x ( 3 − x ) 2 , [ 0 , 2 ]

To determine

To prove: The function, f(x)=427x(3−x)2 is a probability density function over the interval [0,3].

Explanation

Given Information:

The function, f(x)=427x(3−x)2 over the interval [0,3].

Formula used:

Consider a function f of a continuous random variables x whose set of values is the interval [a,b] The function is a probability density function when it is non-negative and continuous on the interval [a,b] and when

∫abf(x)dx=1

Proof:

Consider the function,

f(x)=427x(3−x)2=427x[(3)2−2(3)(x)+(x)2]=427x[9−6x+x2]=427[9x−6x2+x3]

Further simplify the above equation:

f(x)=427⋅9x−427⋅6x2+427⋅x3=43x−89x2+427⋅x3

f(x) is expressible as the sum or difference of different polynomial functions.

All polynomial functions are non-negative and continuous over a bounded interval.

Thus, f(x) is non-negative and continuous over the interval [0,3]

A function f of a continuous random variables x whose set of values is the interval [a,b]. The function is a probability density function when it is non-negative and continuous on the interval [a,b] and when

∫abf(x)dx=1

Now, evaluate ∫abf(x)dx as,

∫03f(x)dx=∫0343xdx−∫0389

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

Chapter 9 Solutions

Show all chapter solutions

Sect-9.1 P-1CP Sect-9.1 P-2CP Sect-9.1 P-3CP Sect-9.1 P-4CP Sect-9.1 P-5CP Sect-9.1 P-6CP Sect-9.1 P-1SWU Sect-9.1 P-2SWU Sect-9.1 P-3SWU Sect-9.1 P-4SWU

Sect-9.1 P-5SWU Sect-9.1 P-6SWU Sect-9.1 P-7SWU Sect-9.1 P-8SWU Sect-9.1 P-9SWU Sect-9.1 P-10SWU Sect-9.1 P-1E Sect-9.1 P-2E Sect-9.1 P-3E Sect-9.1 P-4E Sect-9.1 P-5E Sect-9.1 P-6E Sect-9.1 P-7E Sect-9.1 P-8E Sect-9.1 P-9E Sect-9.1 P-10E Sect-9.1 P-11E Sect-9.1 P-12E Sect-9.1 P-13E Sect-9.1 P-14E Sect-9.1 P-15E Sect-9.1 P-16E Sect-9.1 P-17E Sect-9.1 P-18E Sect-9.1 P-19E Sect-9.1 P-20E Sect-9.1 P-21E Sect-9.1 P-22E Sect-9.1 P-23E Sect-9.1 P-24E Sect-9.1 P-25E Sect-9.1 P-26E Sect-9.1 P-27E Sect-9.1 P-28E Sect-9.1 P-29E Sect-9.1 P-30E Sect-9.1 P-31E Sect-9.1 P-32E Sect-9.1 P-33E Sect-9.1 P-35E Sect-9.1 P-36E Sect-9.1 P-37E Sect-9.1 P-38E Sect-9.2 P-1CP Sect-9.2 P-2CP Sect-9.2 P-3CP Sect-9.2 P-4CP Sect-9.2 P-5CP Sect-9.2 P-1SWU Sect-9.2 P-2SWU Sect-9.2 P-3SWU Sect-9.2 P-4SWU Sect-9.2 P-5SWU Sect-9.2 P-6SWU Sect-9.2 P-7SWU Sect-9.2 P-1E Sect-9.2 P-2E Sect-9.2 P-3E Sect-9.2 P-4E Sect-9.2 P-5E Sect-9.2 P-6E Sect-9.2 P-7E Sect-9.2 P-8E Sect-9.2 P-9E Sect-9.2 P-10E Sect-9.2 P-11E Sect-9.2 P-12E Sect-9.2 P-13E Sect-9.2 P-14E Sect-9.2 P-15E Sect-9.2 P-16E Sect-9.2 P-17E Sect-9.2 P-18E Sect-9.2 P-19E Sect-9.2 P-20E Sect-9.2 P-21E Sect-9.2 P-22E Sect-9.2 P-23E Sect-9.2 P-24E Sect-9.2 P-25E Sect-9.2 P-26E Sect-9.2 P-27E Sect-9.2 P-28E Sect-9.2 P-29E Sect-9.2 P-30E Sect-9.2 P-31E Sect-9.2 P-32E Sect-9.2 P-33E Sect-9.2 P-34E Sect-9.2 P-35E Sect-9.2 P-36E Sect-9.2 P-37E Sect-9.2 P-38E Sect-9.2 P-39E Sect-9.3 P-1CP Sect-9.3 P-2CP Sect-9.3 P-3CP Sect-9.3 P-4CP Sect-9.3 P-5CP Sect-9.3 P-6CP Sect-9.3 P-7CP Sect-9.3 P-1SWU Sect-9.3 P-2SWU Sect-9.3 P-3SWU Sect-9.3 P-4SWU Sect-9.3 P-5SWU Sect-9.3 P-6SWU Sect-9.3 P-1E Sect-9.3 P-2E Sect-9.3 P-3E Sect-9.3 P-4E Sect-9.3 P-5E Sect-9.3 P-6E Sect-9.3 P-7E Sect-9.3 P-8E Sect-9.3 P-9E Sect-9.3 P-10E Sect-9.3 P-11E Sect-9.3 P-12E Sect-9.3 P-13E Sect-9.3 P-14E Sect-9.3 P-15E Sect-9.3 P-16E Sect-9.3 P-17E Sect-9.3 P-18E Sect-9.3 P-19E Sect-9.3 P-20E Sect-9.3 P-21E Sect-9.3 P-22E Sect-9.3 P-23E Sect-9.3 P-24E Sect-9.3 P-25E Sect-9.3 P-26E Sect-9.3 P-27E Sect-9.3 P-28E Sect-9.3 P-29E Sect-9.3 P-30E Sect-9.3 P-31E Sect-9.3 P-32E Sect-9.3 P-33E Sect-9.3 P-34E Sect-9.3 P-35E Sect-9.3 P-36E Sect-9.3 P-37E Sect-9.3 P-38E Sect-9.3 P-39E Sect-9.3 P-40E Sect-9.3 P-41E Sect-9.3 P-42E Sect-9.3 P-43E Sect-9.3 P-44E Sect-9.3 P-45E Sect-9.3 P-46E Sect-9.3 P-47E Sect-9.3 P-48E Sect-9.3 P-49E Sect-9.3 P-50E Sect-9.3 P-54E Sect-9.3 P-55E Sect-9.3 P-56E Sect-9.3 P-57E Ch-9 P-1RE Ch-9 P-2RE Ch-9 P-3RE Ch-9 P-4RE Ch-9 P-5RE Ch-9 P-6RE Ch-9 P-7RE Ch-9 P-8RE Ch-9 P-9RE Ch-9 P-10RE Ch-9 P-11RE Ch-9 P-12RE Ch-9 P-13RE Ch-9 P-14RE Ch-9 P-15RE Ch-9 P-16RE Ch-9 P-17RE Ch-9 P-18RE Ch-9 P-19RE Ch-9 P-20RE Ch-9 P-21RE Ch-9 P-22RE Ch-9 P-23RE Ch-9 P-24RE Ch-9 P-25RE Ch-9 P-26RE Ch-9 P-27RE Ch-9 P-28RE Ch-9 P-29RE Ch-9 P-30RE Ch-9 P-31RE Ch-9 P-32RE Ch-9 P-33RE Ch-9 P-34RE Ch-9 P-35RE Ch-9 P-36RE Ch-9 P-37RE Ch-9 P-38RE Ch-9 P-39RE Ch-9 P-40RE Ch-9 P-41RE Ch-9 P-42RE Ch-9 P-43RE Ch-9 P-44RE Ch-9 P-45RE Ch-9 P-46RE Ch-9 P-47RE Ch-9 P-48RE Ch-9 P-49RE Ch-9 P-50RE Ch-9 P-51RE Ch-9 P-52RE Ch-9 P-53RE Ch-9 P-54RE Ch-9 P-55RE Ch-9 P-56RE Ch-9 P-57RE Ch-9 P-58RE Ch-9 P-59RE Ch-9 P-60RE Ch-9 P-61RE Ch-9 P-62RE Ch-9 P-1TYS Ch-9 P-2TYS Ch-9 P-3TYS Ch-9 P-4TYS Ch-9 P-5TYS Ch-9 P-6TYS Ch-9 P-7TYS Ch-9 P-8TYS Ch-9 P-9TYS Ch-9 P-10TYS Ch-9 P-11TYS Ch-9 P-12TYS Ch-9 P-13TYS Ch-9 P-14TYS Ch-9 P-15TYS Ch-9 P-16TYS

Additional Math Solutions

Find more solutions based on key concepts

Show solutions

MathCalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

Calculus: An Applied Approach (Min...

Solutions

Calculus: An Applied Approach (Min...

Verifying a Probability Density Function In Exercises 1-6, show that the function is a probability density function over the given interval. See Examples 1 and 2. f ( x ) = 4 27 x ( 3 − x ) 2 , [ 0 , 2 ]

Still sussing out bartleby?

The Solution to Your Study Problems

Chapter 9 Solutions

Additional Math Solutions

Find more solutions based on key concepts

Convert each expression in Exercises 25-50 into its technology formula equivalent as in the table in the text. ...

The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), a...

In Exercises 4756, solve the given equation for the indicated variable. 81=3x

A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measure...

Linear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the sol...

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or gi...

Writing an Equation in General Form In Exercises 53-56, use the result of Exercise 52 to write an equation of t...

PA The city manager has received a complaint from the local union of firefighters to the effect that they are u...

Weinstein, McDermott, and Roediger (2010) published an experimental study examining different techniques that s...

28-36. Identify and sketch the graph of each surface. 4x2+y24z2=4

Suppose a line L has equation Ax + By + C = 0. a. What is the slope of L if B 0? b. What is the slope of L if ...

Research has shown that IQ scores have been increasing for years (Flynn, 1984. 1999). The phenomenon is called ...

Statistical Literacy A data set has values ranging from a low of 10 to a high of 50. What's wrong with using th...

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your...

In Exercises 71 and 72, find each of the following, where K, and c are transfinite cardinal numbers. 0+(0+0) 72...

For Problems 55-94, simplify each numerical expression. Objectives 7 4512(35)

In problems 23-58, perform the indicated operations and simplify. 32.

Convert the expressions in Exercises 6584 to power form. xy23

The article A Probabilistic Model of Fracture in Concrete and Size Effects on Fracture Toughness(Magazine of Co...

Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must 02f(x)dx lie?...

Find the amount of sheathing needed for the roof in Illustration 2. How many squares of shingles must be purcha...

In Exercises and, the given table defines an operation of multiplication on the set. In each case, find a cond...

Evaluate each of the following if x is /2 and y is /6. sinx+siny

For Exercises 21 t 24, let P={parallelograms},R={rectangles},andH={rhombi}. Classify as true or false: RH=PandR...

Solve the following problems and reduce to lowest terms. 13.

Average ValueIn Exercises 63-66, find the average value of the function over the given solid region. The averag...

Using Newton's Method In Exercises 7-16, use Newtons Method to approximate the zero(s) or the function. Continu...

The drive mechanism on a treadmill is powered by an electric motor. In the figure, find mD if mABC is 36 larger...

In Exercises 7-16, determine whether the given simplex tableau is in final form. If so, find the solution to th...

Finding Points of Inflection In Exercises 17-32, find the points of inflection and discuss the concavity of the...

Using a binomial series, the Maclaurin series for is:

Evaluating a Line Integral In Exercises 27-30, evaluate the line integral along the given path(s). c(x2+y2)dsC:...

The cylindrical coordinates of the point with rectangular coordinates (5, 5, 4) are:

Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y=12...

For each of the following problems, substitute the given values in the formula and solve for the unknown. Check...

Describe three specific applications of the factorial design: (1) adding a factor to an existing study; (2) usi...

In Exercises 27-56, use a calculator to find each value. Give the entire display of your calculator. a. eln2.3 ...

Applications Use a calculator to help solve each problem. Thorium decay An isotope of thorium, 227Th has a half...

In this chapter, we introduced the observational research design, the survey research design, and the case stud...

For its Music 360 survey, Nielsen Co. asked teenagers and adults how each group has listenedto music in the pas...

To estimate the mean age for a population of 4000 employees, a simple random sample of 40 employees is selected...

Reduced visual performance with increasing age has been a much-studied phenomenon in recent years. This decline...

Math
Calculus Calculus: An Applied Approach (MindTap Course List)10th Edition

Verifying a Probability Density Function In Exercises 1-6, show that the function is a probability density function over the given interval. See Examples 1 and 2.

f ( x ) = 4 27 x ( 3 − x ) 2 , [ 0 , 2 ]