Explanation Given information: Given that f ( x ) be a continuous function defined for 0 ≤ x ≤ 1 . Consider the functions B n ( x ) = ∑ k = 0 n f ( k n ) ( n k ) x k ( 1 − x ) n − k . Calculation: Consider the question, Let X 1 , X 2 .... be the independent Bernoulli random variables with mean x . From the given information, the function can be written as B n ( x ) = ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k f ( k n ) Now consider, ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k = [ x + ( 1 − x ) ] n = 1 ...... ( 1 ) Differentiate the equation ( 1 ) with respect x . ∑ k = 0 n ( n k ) [ k x k − 1 ( 1 − x ) n − k − ( n − k ) x k ( 1 − x ) n − k − 1 ] = 0 Further solving, ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k [ k x − ( n − k ) 1 − x ] = 0 Take the LCM, ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k [ k − n x x ( 1 − x ) ] = 0 Further solving, ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k ( k − n x ) = 0 ...... ( 2 ) Differentiate the equation with respect to x . d d x [ ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k ( k − n x ) ] = 0 Further solving ∑ k = 0 n ( n k ) [ d d x x k ( 1 − x ) n − k ( k − n x ) ] = 0 Solving the term ∑ k = 0 n ( n k ) x k − 1 ( 1 − x ) n − k − 1 [ ( k − n x ) 2 + n x ( x − 1 ) ] = 0 Simplify the function ∑ k = 0 n ( n k ) x k − 1 ( 1 − x ) n − k − 1 ( k − n x ) 2 + ∑ k = 0 n ( n k ) x k + 1 ( 1 − x ) n − k − 1 n x ( x − 1 ) = 0 Further solving ∑ k = 0 n ( n k ) x k − 1 ( 1 − x ) n − k − 1 ( k − n x ) 2 − ∑ k = 0 n ( n k ) x k − 1 + 1 ( 1 − x ) n − k − 1 + 1 n = 0 Solving the values ∑ k = 0 n ( n k ) x k − 1 ( 1 − x ) n − k − 1 ( k − n x ) 2 − n ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k = 0 Multiplication ∑ k = 0 n ( n k ) x k − 1 ( 1 − x ) n − k − 1 ( k − n x ) 2 − n × 1 = 0 Take n to the right side ∑ k = 0 n ( n k ) x k − 1 ( 1 − x ) n − k − 1 ( k − n x ) 2 = n Solving the term Now multiply the equation by x ( 1 − x ) , ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k ( k − n x ) 2 = n x ( 1 − x ) Solving the equation ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k ( x − k n ) 2 = x ( 1 − x ) n

9th Edition

ISBN: 9780321794772

Author: Sheldon Ross

Publisher: PEARSON

See similar textbooks

Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Videos

Question

Chapter 8, Problem 8.5TE

To determine

To find: The function limn→∞Bn(x)=f(x) .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8, Problem 8.4TE

Chapter 8, Problem 8.6TE

Chapter 8 Solutions

A First Course in Probability

Ch. 8 - Suppose that X is a random variable with mean and...Ch. 8 - From past experience, a professor knows that the...Ch. 8 - Use the central limit theorem to solve part (c) of...Ch. 8 - Let X1,...,X20 be independent Poisson random...Ch. 8 - Fifty numbers are rounded off to the nearest...Ch. 8 - A die is continually rolled until the total sum of...Ch. 8 - A person has 100 light bulbs whose lifetimes are...Ch. 8 - In Problem 8.7, suppose that it takes a random...Ch. 8 - If X is a gamma random variable with parameters...Ch. 8 - Civil engineers believe that W, the amount of...

Ch. 8 - Many people believe that the daily change of price...Ch. 8 - We have 100 components that we will put in use in...Ch. 8 - Student scores on exams given by a certain...Ch. 8 - A certain component is critical to the operation...Ch. 8 - An insurance company has 10.000 automobile...Ch. 8 - A.J. has 20 jobs that she must do in sequence,...Ch. 8 - Redo Example 5b under the assumption that the...Ch. 8 - Repeat part (a) of Problem 8.2 when it is known...Ch. 8 - A lake contains 4 distinct types of fish. Suppose...Ch. 8 - If X is a nonne9ative random variable with mean...Ch. 8 - Let X be a nonnegative random variable. Prove that...Ch. 8 - Prob. 8.22P Ch. 8 - Let X be a Poisson random variable with mean 20....Ch. 8 - Prob. 8.24P Ch. 8 - If X has variance 2, then , the positive square...Ch. 8 - If X has mean and standard deviation , the ratio...Ch. 8 - Compute the measurement signal-to-noise ratio-that...Ch. 8 - Let Zn,n1, be a sequence of random variables and...Ch. 8 - Prob. 8.5TE Ch. 8 - Prob. 8.6TE Ch. 8 - Prob. 8.7TE Ch. 8 - Explain why a gamma random variable with...Ch. 8 - Prob. 8.9TE Ch. 8 - If X is a Poisson random variable with mean , show...Ch. 8 - Prob. 8.11TE Ch. 8 - Prob. 8.12TE Ch. 8 - Prob. 8.13TE Ch. 8 - Prob. 8.14TE Ch. 8 - The number of automobiles sold weekly at a certain...Ch. 8 - Prob. 8.2STPE Ch. 8 - If E[X]=75E[Y]=75Var(X)=10var(Y)=12cov(X,Y)=3 give...Ch. 8 - Prob. 8.4STPE Ch. 8 - Prob. 8.5STPE Ch. 8 - Prob. 8.6STPE Ch. 8 - Prob. 8.7STPE Ch. 8 - Prob. 8.8STPE Ch. 8 - Prob. 8.9STPE Ch. 8 - A tobacco company claims that the amount of...Ch. 8 - Prob. 8.11STPE Ch. 8 - Prob. 8.12STPE Ch. 8 - The strong law of large numbers states that with...Ch. 8 - Each new book donated to a library must be...

Knowledge Booster

$Probability$

Learn more about

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

Similar questions

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.
What does it mean to say that f(4) is a local maximum valueof f ?

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
College Algebra
Algebra
ISBN:9781938168383
Author:Jay Abramson
Publisher:OpenStax
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Big Ideas Math A Bridge To Success Algebra 1: Stu...
Algebra
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:Houghton Mifflin Harcourt
Algebra: Structure And Method, Book 1
Algebra
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:McDougal Littell
Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781337278461
Author:Ron Larson
Publisher:Cengage Learning