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(Fuzzy logic.) In fuzzy logic, a proposition (a claim that something is true) has, instead of a binary truth value (0 or 1), a real-valued truth value in the interval [0, 1]. Researchers are modelling a particular proposition's truth value with the random variable X that has density f(x) = [cx (1-x), 10, 0 ≤ x ≤ 1, otherwise. (a) Determine the constant c. The density must integrate to 1. (b) Determine the cumulative distribution function of X, and draw it. (c) Calculate the probability that the truth value is at least 0.75. (d) Find the mode of the distribution, that is, the point x where the density f(x) attains its maximum.

(Fuzzy logic.) In fuzzy logic, a proposition (a claim that something is true) has, instead of a binary truth value (0 or 1), a real-valued truth value in the interval [0, 1]. Researchers are modelling a particular proposition's truth value with the random variable X that has density f(x) = [cx (1-x), 10, 0 ≤ x ≤ 1, otherwise. (a) Determine the constant c. The density must integrate to 1. (b) Determine the cumulative distribution function of X, and draw it. (c) Calculate the probability that the truth value is at least 0.75. (d) Find the mode of the distribution, that is, the point x where the density f(x) attains its maximum.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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(Fuzzy logic.) In fuzzy logic, a proposition (a claim that something is true) has, instead of a binary truth value (0 or 1), a real-valued truth value in the interval [0, 1]. Researchers are modelling a particular proposition's truth value with the random variable X that has density f(x) = cx(1 — x), 0, 0 ≤ x ≤ 1, otherwise. (a) Determine the constant c. The density must integrate to 1. (b) Determine the cumulative distribution function of X, and draw it. (c) Calculate probability that the ruth lue is at least 75. (d) Find the mode of the distribution, that is, the point x where the density f(x) attains its maximum.
Transcribed Image Text:(Fuzzy logic.) In fuzzy logic, a proposition (a claim that something is true) has, instead of a binary truth value (0 or 1), a real-valued truth value in the interval [0, 1]. Researchers are modelling a particular proposition's truth value with the random variable X that has density f(x) = cx(1 — x), 0, 0 ≤ x ≤ 1, otherwise. (a) Determine the constant c. The density must integrate to 1. (b) Determine the cumulative distribution function of X, and draw it. (c) Calculate probability that the ruth lue is at least 75. (d) Find the mode of the distribution, that is, the point x where the density f(x) attains its maximum.
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