Math Problem Statement
Solution
To solve the problem:
-
Find the constant so that is a valid probability density function. The condition for a probability density function is that the total integral over the given range must equal 1. The function given is:
on the interval . We need to compute:
This will allow us to solve for .
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Compute by integrating the probability density function over the interval from 0 to 1:
Step 1: Find
First, compute the integral:
This is a standard integral. Using the substitution , we can simplify the integration:
The integral of is:
Evaluating the limits at and :
Thus:
Step 2: Compute
Now we compute the probability for :
Using the same substitution , the limits change from to , and we have:
Again, the integral of is , so:
Simplifying:
Final Answers:
Would you like further details or have any questions? Here are some follow-up questions to deepen your understanding:
- What is the significance of the constant in a probability density function?
- How does substitution simplify the integration of functions involving transformations like ?
- How do the properties of definite integrals ensure the probability is valid over infinite intervals?
- Could you extend the calculation for different intervals, say ?
- Why is it important that the total probability over the interval equals 1?
Tip: When dealing with probability density functions, always ensure the integral over the entire range equals 1 to validate the function.
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Math Problem Analysis
Mathematical Concepts
Probability Density Function
Integration
Substitution Method
Formulas
p(x) = C / (x + 4)^3
∫_0^∞ (C / (x+4)^3) dx = 1
P(0 ≤ X ≤ 1) = ∫_0^1 (C / (x+4)^3) dx
Theorems
Definite Integrals
Properties of Probability Density Functions
Suitable Grade Level
Undergraduate
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