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Suppose a probability density function is given by f (x) = Cxe; 0 0 for all r to be 0 there.)

Suppose a probability density function is given by f (x) = Cxe; 0 0 for all r to be 0 there.)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Step (b): Utilizing the Result from Part (a)** **Objective:** To deduce the value of constant C based on the findings from Part (a). **Instruction:** Then, from the result of part (a), deduce the value of C. **Reference:** Page 5 --- In this educational resource, learners are expected to derive the value of the constant C using the results obtained from their previous computations or results in Part (a). This step follows directly from the preceding calculations and logical reasoning established earlier in the document. For better understanding and to ensure accurate deductions, students are encouraged to meticulously revisit their results from Part (a), verify the accuracy of their findings, and apply appropriate mathematical or scientific principles. --- This task does not contain any graphs or diagrams that need explanations. Therefore, focus mainly remains on the textual instructions provided.
Transcribed Image Text:**Step (b): Utilizing the Result from Part (a)** **Objective:** To deduce the value of constant C based on the findings from Part (a). **Instruction:** Then, from the result of part (a), deduce the value of C. **Reference:** Page 5 --- In this educational resource, learners are expected to derive the value of the constant C using the results obtained from their previous computations or results in Part (a). This step follows directly from the preceding calculations and logical reasoning established earlier in the document. For better understanding and to ensure accurate deductions, students are encouraged to meticulously revisit their results from Part (a), verify the accuracy of their findings, and apply appropriate mathematical or scientific principles. --- This task does not contain any graphs or diagrams that need explanations. Therefore, focus mainly remains on the textual instructions provided.
### Probability Density Function Task Suppose a probability density function is given by: \[ f(x) = Cxe^{-x^2} \, \text{for} \, 0 < x < \infty \] and \( f(x) = 0 \) otherwise. Since \( f(x) \) is only defined on \( 0 < x < \infty \), it must satisfy the properties: 1. \( f(x) \geq 0 \) for all \( x \) 2. \[ \int_0^\infty f(x) \, dx = 1 \] (We do not consider \(( -\infty, 0]\) since \( f(x) \) is defined to be 0 there). With these properties in mind, determine the value of \( C \) by the following. ### Task (a) First, prove: \[ \int_0^\infty xe^{-x^2} \, dx = \frac{1}{2} \] --- This task requires knowledge of probability density functions and integral calculus to solve. The main objective is to determine the normalization constant \( C \) so that the given function \( f(x) \) satisfies the properties required of a probability density function over the specified domain.
Transcribed Image Text:### Probability Density Function Task Suppose a probability density function is given by: \[ f(x) = Cxe^{-x^2} \, \text{for} \, 0 < x < \infty \] and \( f(x) = 0 \) otherwise. Since \( f(x) \) is only defined on \( 0 < x < \infty \), it must satisfy the properties: 1. \( f(x) \geq 0 \) for all \( x \) 2. \[ \int_0^\infty f(x) \, dx = 1 \] (We do not consider \(( -\infty, 0]\) since \( f(x) \) is defined to be 0 there). With these properties in mind, determine the value of \( C \) by the following. ### Task (a) First, prove: \[ \int_0^\infty xe^{-x^2} \, dx = \frac{1}{2} \] --- This task requires knowledge of probability density functions and integral calculus to solve. The main objective is to determine the normalization constant \( C \) so that the given function \( f(x) \) satisfies the properties required of a probability density function over the specified domain.
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