**Title: Understanding Probability Density Functions in Mail-Order Solicitations****Introduction:**Learn how to calculate the expected proportion of individuals who respond to a mail-order solicitation using probability density functions. **Problem Statement:**Find the proportion $X$ of individuals expected to respond to a certain mail-order solicitation if $X$ has the density function defined below.**Density Function:**\[ f(x) = \begin{cases} \frac{2(x+4)}{9}, & 0 < x < 1, \\0, & \text{elsewhere.}\end{cases} \]**Explanation:**- **Range of $x$:** The function is defined for $x$ between 0 and 1, representing possible values of the proportion $X$ that respond.- **Elsewhere Condition:** The probability density is 0 for values of $x$ outside the interval (0,1).**Objective:**Calculate the proportion of individuals responding using this defined density function. Integrate $f(x)$ over its range to find the probability. ### Problem Statement:If a dealer's profit, in units of $3000, on a new automobile can be viewed as a random variable $ X $ with the density function given below, find the average profit per automobile.### Density Function:\[ f(x) = \begin{cases} \frac{1}{12}(7 - x), & 0 < x < 2, \\0, & \text{elsewhere}\end{cases} \]### Explanation:The function $ f(x) $ is defined as follows:- Within the interval $ 0 < x < 2 $, the probability density function is $ \frac{1}{12}(7 - x) $.- For values of $ x $ outside this interval, the density function is 0, indicating that profit values outside this range are not considered in this model.To find the average (or expected) profit per automobile, you need to calculate the expected value $ E(X) $ of the random variable $ X $ using the density function provided. This involves integrating the product of $ x $ and $ f(x) $ over the interval from 0 to 2:\[ E(X) = \int_0^2 x \cdot f(x) \, dx \]The expected value will give the average profit in units of $3000. Multiply the result by $3000 to find the average profit in dollars.

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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An important factor in solid missile fuel is the particle size distribution. Significant problems occur if the particle sizes are too large. From production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by the following function. 7x8, x>1 f(x) = elsewhere (a) Verify that this is a valid density function. (b) Evaluate F(x). (c) What is the probability that a random particle from the manufactured fuel exceeds 3 micrometers?
An important factor in solid missile fuel is the particle size distribution. Significant problems occur if the particle sizes are too large. From production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by the following function. -8 7x x>1 f(x) = 0₁ elsewhere (a) Verify that this is a valid density function. (b) Evaluate F(x). (c) What is the probability that a random particle from the manufactured fuel exceeds 2 micrometers? C (a) The function f(x) is a valid density function because ▼ for all and because S f(x) dx = = √ 7x²³dx = ₁₁ = - 00 x≤ 1, (b) F(x)= 48 x> 1. (c) P(X> 2) = (Type an integer or decimal rounded to four decimal places as needed.)
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Xavier and Yolanda both have classes that end at noon and they agree to meet every day after class. They arrive at the coffee shop independently. Xavier's arrival time is X and Yolanda's arrival time is Y, where X and Y are measured in minutes after noon. The individual density functions are given.(Xavier arrives sometime after noon and is more likely to arrive promptly than late. Yolanda always arrives by 12:10 PM and is more likely to arrive late than promptly.) After Yolanda arrives, she'll wait for up to 45 minutes for Xavier, but he won't wait for her. Find the probability that they meet. (Round your answer to three decimal places.)

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Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function below. f(x) = 2(x+4) 9 0₁ 0