Math Problem Statement
Solution
To solve this problem, let's go step by step for each part:
a) Find so that is a valid probability density function
For to be a valid probability density function (PDF), the integral over its range must equal 1:
The integral of is:
Evaluate the definite integral:
Set the integral equal to 1:
Solve for :
b) Find the probability that
Since this is a continuous distribution, the probability that is:
Substitute :
c) Find the probability that
We calculate by integrating from 3 to 4:
The integral of is:
Substitute :
d) What is the mean of ?
The mean is given by:
The integral of is:
Evaluate the definite integral:
e) What is the variance of ?
The variance is:
[ \sigma^2 = E[X^2] - (E[X])^2 ]
First, calculate :
The integral of is:
Evaluate the definite integral:
[ E[X^2] = \frac{1}{k} \left[ \frac{x^8}{8} \right]_0^4 = \frac{1}{682.6667} \cdot \frac{4^8}{8} ]
Now calculate the variance:
[ \sigma^2 = E[X^2] - (E[X])^2 = 12.0000 - (3.4230)^2 = 12.0000 - 11.7151 = 0.2849 ]
Final Answers:
a)
b)
c)
d) Mean () = 3.4230
e) Variance () = 0.2849
Would you like further clarification or any specific part re-explained? Here are 5 follow-up questions:
- How is the normalization constant calculated for a probability density function?
- Why is the probability at a single point zero for a continuous distribution?
- How can the cumulative distribution function (CDF) help with these calculations?
- What is the relationship between and variance in a distribution?
- How would the answers change if the range of was different?
Tip: Always check that a PDF integrates to 1 over its range to ensure it's valid!
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Math Problem Analysis
Mathematical Concepts
Probability Density Functions
Definite Integration
Mean and Variance of Continuous Random Variables
Formulas
Integral of PDF over range = 1: ∫[0,4] f(x) dx = 1
Mean: μ = ∫[0,4] x * f(x) dx
Variance: σ^2 = E[X^2] - (E[X])^2
Theorems
Normalization condition for a valid PDF
Calculation of expectation and variance for continuous random variables
Suitable Grade Level
University Level - Probability and Statistics
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