Math Problem Statement
Solution
To solve the given problem, we will follow these steps systematically:
Step 1: Normalize
To ensure is a probability density function, it must satisfy:
We compute:
Since is symmetric around , the integral becomes:
The integral of from 0 to is 1, so:
Thus, the probability density function is:
Step 2: Find the Mean ()
The mean of a probability density function is:
Due to symmetry, the mean of is 0. The positive and negative contributions to the integral cancel out:
Step 3: Find the Variance ()
The variance is given by:
Substituting , we have:
Using symmetry, we evaluate only the integral from to and double it:
To compute , we use the gamma function ( for integers):
Thus:
Step 4: Find
To compute , we integrate from 0 to 4:
The integral of is , so:
Final Results:
- Normalization constant ():
- Mean ():
- Variance ():
- Probability :
Would you like detailed steps on how to calculate integrals, or do you have additional questions?
Expanding Questions:
- How does the symmetry of a function simplify calculations for mean and variance?
- What is the relationship between gamma functions and factorials?
- Can we generalize this result for other exponential functions?
- What does represent in terms of probability?
- Why does represent a valid probability distribution?
Tip:
When dealing with symmetric distributions, always check if symmetry simplifies the calculation of expected value or variance!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability Density Function
Normalization
Symmetry
Expected Value
Variance
Exponential Distribution
Formulas
Integral for normalization: ∫_{-∞}^∞ f(x) dx = 1
Mean: μ = ∫_{-∞}^∞ x f(x) dx
Variance: σ² = ∫_{-∞}^∞ x² f(x) dx - μ²
Probability: P(a < X < b) = ∫_{a}^b f(x) dx
Theorems
Gamma Function Relation to Factorials
Properties of Symmetric Functions in Probability
Suitable Grade Level
Undergraduate
Related Recommendation
Solve for k and Mean of Random Variable X with Exponential PDF
Determine K and Calculate Mean for Gaussian-like PDF
Probability Density Function and Moment Generating Function: Finding k and Mean of Random Variable X
Continuous Random Variable PDF with Normalization
Calculating Expected Value and Variance for Exponential Distribution f(x) = 4e^(-4x)