Explanation Given: The function is f ( x ) = k x 3 2 on the interval [ 4 , 9 ] . Definition used: Probability density function : “The function f is a probability density function of a random variable X in the interval [ a , b ] if 1. f ( x ) ≥ 0 for all x in the interval [ a , b ] . 2. ∫ a b f ( x ) d x = 1 ” Cumulative distribution function : “If f is a probability density function of a random variable in the interval [ − a , b ] , then the Cumulative distribution function is defined as F ( x ) = P ( X ≤ x ) = ∫ a x f ( t ) d t for x ≥ a . Also F ( x ) = 0 for x < a ”. Calculation: Consider the given function f ( x ) = k x 3 2 . Integrate the function f ( x ) = k x 3 2 on the interval [ 4 , 9 ] . ∫ 4 9 f ( x ) d x = ∫ 4 9 k x 3 2 d x = k ∫ 4 9 x 3 2 d x = k [ x 3 2 + 1 ( 3 2 + 1 ) ] 4 9 ( ∵ ∫ x n d x = x n + 1 n + 1 + C ) = 2 k 5 [ x 5 2 ] 4 9 On further simplification, ∫ 4 9 f ( x ) d x = 2 k 5 [ 9 5 2 − 4 5 2 ] = 2 k 5 [ ( 3 2 ) 5 2 − ( 2 2 ) 5 2 ] = 2 k 5 ( 3 5 − 2 5 ) = 2 k 5 ( 243 − 32 ) = 422 k 5 By the definition, it is observed that the integral must be equal to 1 for the function to be a probability density function

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 18.1, Problem 24E

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To find: The cumulative distribution function for the probability density function.

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Chapter 18.1, Problem 23E

Chapter 18.1, Problem 25E

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Chapter 18 Solutions

Finite Mathematics and Calculus with Applications

Ch. 18.1 - Prob. 1YT Ch. 18.1 - Prob. 2YT Ch. 18.1 - Using the probability density function of Example...Ch. 18.1 - Use part (a) of Example 5 to calculate the...Ch. 18.1 - Evaluate each of the following integrals. (Sec....Ch. 18.1 - Prob. 2WE Ch. 18.1 - Prob. 3WE Ch. 18.1 - Decide whether the functions defined as follows...Ch. 18.1 - Prob. 2E Ch. 18.1 - Prob. 3E

Ch. 18.1 - Prob. 4E Ch. 18.1 - Prob. 5E Ch. 18.1 - Prob. 6E Ch. 18.1 - Prob. 7E Ch. 18.1 - Prob. 8E Ch. 18.1 - Prob. 9E Ch. 18.1 - Prob. 10E Ch. 18.1 - Prob. 11E Ch. 18.1 - Prob. 12E Ch. 18.1 - Prob. 13E Ch. 18.1 - Prob. 14E Ch. 18.1 - Prob. 15E Ch. 18.1 - Prob. 16E Ch. 18.1 - Prob. 17E Ch. 18.1 - Prob. 18E Ch. 18.1 - Prob. 19E Ch. 18.1 - Prob. 20E Ch. 18.1 - Prob. 21E Ch. 18.1 - Find the cumulative distribution function for the...Ch. 18.1 - Prob. 23E Ch. 18.1 - Prob. 24E Ch. 18.1 - Prob. 25E Ch. 18.1 - Prob. 26E Ch. 18.1 - Prob. 27E Ch. 18.1 - Prob. 28E Ch. 18.1 - Prob. 29E Ch. 18.1 - Prob. 30E Ch. 18.1 - Prob. 31E Ch. 18.1 - Prob. 32E Ch. 18.1 - Prob. 33E Ch. 18.1 - Prob. 34E Ch. 18.1 - Life Span of a Computer Part The life (in months)...Ch. 18.1 - Prob. 36E Ch. 18.1 - Prob. 37E Ch. 18.1 - Prob. 38E Ch. 18.1 - Prob. 39E Ch. 18.1 - Prob. 40E Ch. 18.1 - Prob. 41E Ch. 18.1 - Flea Beetles The mobility of an insect is an...Ch. 18.1 - Prob. 43E Ch. 18.1 - Prob. 44E Ch. 18.1 - Prob. 45E Ch. 18.1 - Earthquakes The time between major earthquakes in...Ch. 18.1 - Earthquakes The time between major earthquakes in...Ch. 18.1 - Prob. 48E Ch. 18.1 - Driving Fatalities We saw in a review exercise in...Ch. 18.1 - Prob. 50E Ch. 18.1 - Time of Traffic Fatality The National Highway...Ch. 18.2 - Repeat Example l for the probability density...Ch. 18.2 - Prob. 2YT Ch. 18.2 - Prob. 3YT Ch. 18.2 - Find P(1 X 2) for each probability function on...Ch. 18.2 - Prob. 2WE Ch. 18.2 - Prob. 1E Ch. 18.2 - Prob. 2E Ch. 18.2 - Prob. 3E Ch. 18.2 - Prob. 4E Ch. 18.2 - Prob. 5E Ch. 18.2 - Prob. 6E Ch. 18.2 - Prob. 7E Ch. 18.2 - In Exercises 18, a probability density function of...Ch. 18.2 - Prob. 9E Ch. 18.2 - Prob. 10E Ch. 18.2 - Prob. 11E Ch. 18.2 - Prob. 12E Ch. 18.2 - Prob. 13E Ch. 18.2 - Prob. 14E Ch. 18.2 - Prob. 15E Ch. 18.2 - Prob. 16E Ch. 18.2 - Prob. 17E Ch. 18.2 - For Exercises 1520, (a) find the median of the...Ch. 18.2 - Prob. 19E Ch. 18.2 - Prob. 20E Ch. 18.2 - Prob. 21E Ch. 18.2 - Prob. 22E Ch. 18.2 - Prob. 23E Ch. 18.2 - Prob. 24E Ch. 18.2 - Prob. 25E Ch. 18.2 - Prob. 26E Ch. 18.2 - Losses After Deductible A manufacturers annual...Ch. 18.2 - Prob. 28E Ch. 18.2 - Prob. 29E Ch. 18.2 - Prob. 30E Ch. 18.2 - Prob. 31E Ch. 18.2 - Prob. 32E Ch. 18.2 - Petal Length The length (in centimeters) of a...Ch. 18.2 - Prob. 34E Ch. 18.2 - Prob. 35E Ch. 18.2 - Prob. 36E Ch. 18.2 - Prob. 37E Ch. 18.2 - Prob. 38E Ch. 18.2 - Annual Rainfall The annual rainfall in a remote...Ch. 18.2 - Prob. 40E Ch. 18.2 - Prob. 41E Ch. 18.2 - Prob. 42E Ch. 18.2 - Time of Traffic Fatality In Exercise 51 of the...Ch. 18.3 - Prob. 1YT Ch. 18.3 - Prob. 2YT Ch. 18.3 - Prob. 3YT Ch. 18.3 - Prob. 1WE Ch. 18.3 - Prob. 2WE Ch. 18.3 - Prob. 1E Ch. 18.3 - Prob. 2E Ch. 18.3 - Find (a) the mean of the distribution, (b) the...Ch. 18.3 - Prob. 4E Ch. 18.3 - Prob. 5E Ch. 18.3 - Prob. 6E Ch. 18.3 - Prob. 7E Ch. 18.3 - Prob. 8E Ch. 18.3 - Prob. 9E Ch. 18.3 - Prob. 10E Ch. 18.3 - Prob. 11E Ch. 18.3 - Prob. 12E Ch. 18.3 - Prob. 13E Ch. 18.3 - Prob. 14E Ch. 18.3 - Prob. 15E Ch. 18.3 - Prob. 16E Ch. 18.3 - Prob. 17E Ch. 18.3 - Prob. 18E Ch. 18.3 - Prob. 19E Ch. 18.3 - Prob. 20E Ch. 18.3 - Prob. 21E Ch. 18.3 - Prob. 22E Ch. 18.3 - Prob. 23E Ch. 18.3 - Prob. 24E Ch. 18.3 - Prob. 25E Ch. 18.3 - Prob. 26E Ch. 18.3 - Prob. 27E Ch. 18.3 - Prob. 28E Ch. 18.3 - Prob. 29E Ch. 18.3 - Prob. 30E Ch. 18.3 - Prob. 31E Ch. 18.3 - Prob. 32E Ch. 18.3 - Prob. 33E Ch. 18.3 - Prob. 34E Ch. 18.3 - Insured Loss An insurance policy is written to...Ch. 18.3 - Prob. 36E Ch. 18.3 - Printer Failure The lifetime of a printer costing...Ch. 18.3 - Prob. 38E Ch. 18.3 - Prob. 39E Ch. 18.3 - Prob. 40E Ch. 18.3 - Prob. 41E Ch. 18.3 - Prob. 42E Ch. 18.3 - Finding Prey H. 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Chapter 18 Solutions