Explanation Definition used: Median: It is the value of the random variable and which divides the area under the graph of the probability density function into 2 equal parts. If m satisfy the condition P ( X ≤ m ) = 1 2 , then m is the median. Calculation: The given function is f ( x ) = { 4 x − 4 x 3 if 0 ≤ x ≤ 1 0 othewise . Initially consider the case when x < 0 . Use the definition of the cumulative distribution function as, F ( x ) = P ( X < 0 ) = ∫ − ∞ x 0 d t [ ∵ f ( x ) = 0 for x < 0 ] = 0 Hence, the cumulative distribution function for x < 0 is F ( x ) = 0 . In the same manner obtain the cumulative distribution function when x ≥ 0 . F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t = ∫ − ∞ 0 0 d t + ∫ 0 x ( 4 t − 4 t 3 ) d t On further simplification, F ( x ) = ∫ 0 x ( 4 t − 4 t 3 ) d t = ( 2 t 2 − t 4 ) 0 x = 2 x 2 − x 4 So, the cumulative distribution function for x ≥ 0 is F ( x ) = 2 x 2 − x 4 . Obtain the cumulative distribution function when 0 ≤ x ≤ 1

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

Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could co…

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a…

Videos

Question

Chapter 11.3, Problem 41E

To determine

To find: The median of the probability density function f(x)={4x−4x3 if 0≤x≤10 othewise.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.3, Problem 40E

Chapter 11.3, Problem 42E

Students have asked these similar questions

Construct a histogram for the spot weld shear strength datain Exercise 6.2.9. Comment on the shape of the histogram. Doesit convey the same information as the stem-and-leaf display? Reference: Exercise 6.2.9 is found in the image attached below

1. Show that f(x) = x3 is not uniformly continuous on R. 2. Show that f(x) = 1/(x-2) is not uniformly continuous on (2,00). 3. Show that f(x)=sin(1/x) is not uniformly continuous on (0,л/2]. 4. Show that f(x) = mx + b is uniformly continuous on R. 5. Show that f(x) = 1/x2 is uniformly continuous on [1, 00), but not on (0, 1]. 6. Show that if f is uniformly continuous on [a, b] and uniformly continuous on D (where D is either [b, c] or [b, 00)), then f is uniformly continuous on [a, b]U D. 7. Show that f(x)=√x is uniformly continuous on [1, 00). Use Exercise 6 to conclude that f is uniformly continuous on [0, ∞). 8. Show that if D is bounded and f is uniformly continuous on D, then fis bounded on D. 9. Let f and g be uniformly continuous on D. Show that f+g is uniformly continuous on D. Show, by example, that fg need not be uniformly con- tinuous on D. 10. Complete the proof of Theorem 4.7. 11. Give an example of a continuous function on Q that cannot be continuously extended to R. 12.…

3. Explain why the following statements are not correct. a. "With my methodological approach, I can reduce the Type I error with the given sample information without changing the Type II error." b. "I have already decided how much of the Type I error I am going to allow. A bigger sample will not change either the Type I or Type II error." C. "I can reduce the Type II error by making it difficult to reject the null hypothesis." d. "By making it easy to reject the null hypothesis, I am reducing the Type I error."

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 median of the probability density function f ( x ) = { 4 x − 4 x 3 if 0 ≤ x ≤ 1 0 othewise .

Solution Summary: The author explains the probability density function's median, which is 0.54, and the cumulative distribution function for x0.

Concept explainers

Videos

To find: The median of the probability density function f(x)={4x−4x3 if 0≤x≤10 othewise.

Want to see the full answer?

Chapter 11 Solutions