Math Problem Statement
Find c for the given pdfs
Solution
To determine the value of for the given probability density functions (pdfs), we need to ensure that each function satisfies the properties of a valid pdf. Specifically, the total area under the curve of the pdf must equal 1. This translates mathematically to:
For each case in the image:
(a) for
We need to solve the following integral: Solving this:
(b) for
We need to solve: Solving this:
(c) for
We need to solve: Solving this: First, compute the integral: [ \int_0^2 (2 - x) , dx = \left[ 2x - \frac{x^2}{2} \right]_0^2 = \left[ 4 - 2 \right] = 2 ] So, we get:
Let me know if you'd like further details on any step or additional questions! Here are five related questions for practice:
- What is the general condition for a function to be a valid probability density function?
- How would you normalize a pdf if it’s not normalized already?
- How would the pdf change if the range for (b) was ?
- What happens to the constant in a pdf if the range of integration is doubled?
- How do you interpret the value of in the context of these problems?
Tip: Always ensure that the integral of any pdf over its range equals 1 to satisfy the normalization condition!
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Math Problem Analysis
Mathematical Concepts
Probability Density Functions
Integral Calculus
Normalization Condition
Formulas
∫ f(x) dx = 1 (Normalization condition for a valid pdf)
∫ c x^3 dx
∫ c dx
∫ c (2 - x) dx
Theorems
Fundamental Theorem of Calculus
Normalization Condition for PDFs
Suitable Grade Level
University Level (Advanced Calculus/Probability)
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