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The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following density function. f(x) = {0.075x + (a) Graph the pdf. f(x) 0.6 0.5 0.4 0.3 0.2 0.1 O f(x) 0.6 0.5 0.4 0.3 0.2 0.1 O 1 1 0.07 0.075x + 0.2 dx = 0.2 (b) Calculate P(X ≤ 4). 2 3 ≤x≤ 5 otherwise 3 2 3 = 1.9375 - 4 5 4 5 5 X X f(x) 0.6 0.5 Verify that the total area under the density curve is indeed 1. 0.4 0.3 0.2 0.1 O 1 2 3 4 5 X f(x) 0.6 0.5 0.4 0.3 0.2 0.1 O 1 2 3 4 5 X

The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following density function. f(x) = {0.075x + (a) Graph the pdf. f(x) 0.6 0.5 0.4 0.3 0.2 0.1 O f(x) 0.6 0.5 0.4 0.3 0.2 0.1 O 1 1 0.07 0.075x + 0.2 dx = 0.2 (b) Calculate P(X ≤ 4). 2 3 ≤x≤ 5 otherwise 3 2 3 = 1.9375 - 4 5 4 5 5 X X f(x) 0.6 0.5 Verify that the total area under the density curve is indeed 1. 0.4 0.3 0.2 0.1 O 1 2 3 4 5 X f(x) 0.6 0.5 0.4 0.3 0.2 0.1 O 1 2 3 4 5 X

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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Please solve a, b, and c.

### Educational Explanation of Probability Density Function and Graph Analysis #### Continuous Random Variable and Density Function In the study of probability and statistics, the current in a certain circuit as measured by an ammeter is considered a continuous random variable \(X\). The probability density function (PDF) for \(X\) is defined as: \[ f(x) = \begin{cases} 0.075x + 0.2 & \text{for } 3 \leq x \leq 5 \\ 0 & \text{otherwise} \end{cases} \] This piecewise function specifies that \(f(x)\) is only non-zero within the interval [3, 5]. Outside this range, the probability density is zero. #### Graphical Representation **Task (a): Graph the PDF** Four possible graphs are displayed, each labeled with the function given: 1. **First Graph:** A line segment increasing from \(x = 3\) to \(x = 5\) with endpoints approximately at (3, 0.425) and (5, 0.575). 2. **Second Graph:** A line segment increasing from \(x = 3\) to \(x = 4\) with endpoints approximately at (3, 0.2) and (4, 0.475). 3. **Third Graph:** A line segment decreasing from \(x = 3\) to \(x = 5\) with endpoints approximately at (3, 0.425) and (5, 0.2). 4. **Fourth Graph:** A line segment decreasing from \(x = 3\) to \(x = 5\) ending around the x-axis. **Correct Graph Selection:** The correct graph should represent the formula \(f(x) = 0.075x + 0.2\) between \(x = 3\) and \(x = 5\). #### Verification and Calculation **Verification: Total Area Under the Curve** Verify that the total area under the density curve equals 1 by calculating: \[ \int_{3}^{5} (0.075x + 0.2) \, dx \] Steps: 1. Calculate the integral: - \(\int (0.075x + 0.2) \, dx = \left[0.0375x^2 + 0.2x\right
Transcribed Image Text:### Educational Explanation of Probability Density Function and Graph Analysis #### Continuous Random Variable and Density Function In the study of probability and statistics, the current in a certain circuit as measured by an ammeter is considered a continuous random variable \(X\). The probability density function (PDF) for \(X\) is defined as: \[ f(x) = \begin{cases} 0.075x + 0.2 & \text{for } 3 \leq x \leq 5 \\ 0 & \text{otherwise} \end{cases} \] This piecewise function specifies that \(f(x)\) is only non-zero within the interval [3, 5]. Outside this range, the probability density is zero. #### Graphical Representation **Task (a): Graph the PDF** Four possible graphs are displayed, each labeled with the function given: 1. **First Graph:** A line segment increasing from \(x = 3\) to \(x = 5\) with endpoints approximately at (3, 0.425) and (5, 0.575). 2. **Second Graph:** A line segment increasing from \(x = 3\) to \(x = 4\) with endpoints approximately at (3, 0.2) and (4, 0.475). 3. **Third Graph:** A line segment decreasing from \(x = 3\) to \(x = 5\) with endpoints approximately at (3, 0.425) and (5, 0.2). 4. **Fourth Graph:** A line segment decreasing from \(x = 3\) to \(x = 5\) ending around the x-axis. **Correct Graph Selection:** The correct graph should represent the formula \(f(x) = 0.075x + 0.2\) between \(x = 3\) and \(x = 5\). #### Verification and Calculation **Verification: Total Area Under the Curve** Verify that the total area under the density curve equals 1 by calculating: \[ \int_{3}^{5} (0.075x + 0.2) \, dx \] Steps: 1. Calculate the integral: - \(\int (0.075x + 0.2) \, dx = \left[0.0375x^2 + 0.2x\right
(b) Calculate \( P(X \leq 4) \). [Text Box] How does this probability compare to \( P(X < 4) \)? - \( P(X \leq 4) < P(X < 4) \) - \( P(X \leq 4) > P(X < 4) \) - \( P(X \leq 4) = P(X < 4) \) (c) Calculate \( P(3.5 \leq X \leq 4.5) \). [Text Box] Calculate \( P(4.5 < X) \). [Text Box]
Transcribed Image Text:(b) Calculate \( P(X \leq 4) \). [Text Box] How does this probability compare to \( P(X < 4) \)? - \( P(X \leq 4) < P(X < 4) \) - \( P(X \leq 4) > P(X < 4) \) - \( P(X \leq 4) = P(X < 4) \) (c) Calculate \( P(3.5 \leq X \leq 4.5) \). [Text Box] Calculate \( P(4.5 < X) \). [Text Box]
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