Explanation Definition used: The function f ( x ) is a probability density function for a continuous random variable X if, 1. f ( x ) ≥ 0 , for all x ∈ ( − ∞ , ∞ ) 2. ∫ − ∞ ∞ f ( x ) d x = 1 3. The probability that X lies in the interval [ c , d ] is given by P ( c ≤ X ≤ d ) = ∫ c d f ( x ) d x . Calculation: The given function is, f ( x ) = 1 2 π e − 0.5 x 2 . Use online graphing calculator and sketch the graph of the function f ( x ) as shown below in Figure 1. From Figure 1, it is observed that, the maximum value of the function is approximately at 0.399. Notice from the graph that the maximum value occurs at the point x = 0 . Obtain the value of ∫ − N N f ( x ) d x for N = 1 , 2 and 3 as follows. For N = 1 , the integral becomes ∫ − 1 1 1 2 π e − 0.5 x 2 d x . Compute the value of ∫ − 1 1 1 2 π e − 0.5 x 2 d x as follows. ∫ − 1 1 1 2 π e − 0.5 x 2 d x = 1 2 π ∫ − 1 1 e − 0.5 x 2 d x Use online calculator to find the integral value of ∫ − 1 1 e − 0.5 x 2 d x . From the above screenshot, it is observed that the value of the integral ∫ − 1 1 e − 0.5 x 2 d x is approximately equal to 1.71125 . Substitute ∫ − 1 1 e − 0.5 x 2 d x ≈ 1.71125 in 1 2 π ∫ − 1 1 e − 0.5 x 2 d x as, 1 2 π ∫ − 1 1 e − 0.5 x 2 d x ≈ 1 ( 2 ) 3.1415 ( 1.71125 ) ≈ 1 6.283 ( 1.71125 ) ≈ 1.71125 2.5066 ≈ 0.683 Thus, the integral ∫ − 1 1 1 2 π e − 0.5 x 2 d x is approximately equal to 0.683. Calculate the value of ∫ − 2 2 1 2 π e − 0.5 x 2 d x as follows. ∫ − 2 2 1 2 π e − 0.5 x 2 d x = 1 2 π ∫ − 2 2 e − 0.5 x 2 d x Use online calculator to calculate the integral value of ∫ − 2 2 e − 0.5 x 2 d x . From the above screenshot, it is observed that the value of the integral ∫ − 2 2 e − 0.5 x 2 d x is approximately equal to 2.39258 . Substitute ∫ − 2 2 e − 0.5 x 2 d x ≈ 2.39258 in 1 2 π ∫ − 2 2 e − 0.5 x 2 d x as, 1 2 π ∫ − 2 2 e − 0.5 x 2 d x ≈ 1 ( 2 ) 3.1415 ( 2.39258 ) ≈ 1 6.283 ( 2.39258 ) ≈ 2.39258 2.5066 ≈ 0.955 Thus, the integral ∫ − 2 2 1 2 π e − 0

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 11.2, Problem 47E

To determine

To sketch: The graph of the function f(x); compute the integral ∫−NNf(x)dx for N=1,2 and 3 and also explain the proof for f being a probability density function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.2, Problem 46E

Chapter 11.2, Problem 48E

Students have asked these similar questions

Questions An insurance company's cumulative incurred claims for the last 5 accident years are given in the following table: Development Year Accident Year 0 2018 1 2 3 4 245 267 274 289 292 2019 255 276 288 294 2020 265 283 292 2021 263 278 2022 271 It can be assumed that claims are fully run off after 4 years. The premiums received for each year are: Accident Year Premium 2018 306 2019 312 2020 318 2021 326 2022 330 You do not need to make any allowance for inflation. 1. (a) Calculate the reserve at the end of 2022 using the basic chain ladder method. (b) Calculate the reserve at the end of 2022 using the Bornhuetter-Ferguson method. 2. Comment on the differences in the reserves produced by the methods in Part 1.

Answer questions 8.1.13 and 8.2.10 respectively

To help consumers in purchasing a laptop computer, Consumer Reports calculates an overall test score for each computer tested based upon rating factors such as ergonomics, portability, performance, display, and battery life. Higher overall scores indicate better test results. The following data show the average retail price and the overall score for ten 13-inch models (Consumer Reports website, October 25, 2012). Brand & Model Price ($) Overall Score Samsung Ultrabook NP900X3C-A01US 1250 83 Apple MacBook Air MC965LL/A 1300 83 Apple MacBook Air MD231LL/A 1200 82 HP ENVY 13-2050nr Spectre XT 950 79 Sony VAIO SVS13112FXB 800 77 Acer Aspire S5-391-9880 Ultrabook 1200 74 Apple MacBook Pro MD101LL/A 1200 74 Apple MacBook Pro MD313LL/A 1000 73 Dell Inspiron I13Z-6591SLV 700 67 Samsung NP535U3C-A01US 600 63 a. Select a scatter diagram with price as the independent variable. b. What does the scatter diagram developed in part (a) indicate about the relationship…

Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The graph of the function f ( x ) ; compute the integral ∫ − N N f ( x ) d x for N = 1 , 2 and 3 and also explain the proof for f being a probability density function.

Solution Summary: The author illustrates the graph of the function f(x) using an online graphing calculator.

Videos

To sketch: The graph of the function f(x); compute the integral ∫−NNf(x)dx for N=1,2 and 3 and also explain the proof for f being a probability density function.

Want to see the full answer?

Chapter 11 Solutions