Math Problem Statement
If X is uniform over (0,1), calculate E(Xn) and Var(Xn)
Solution
Given that is uniformly distributed over , and we want to calculate and , we can proceed as follows:
1. Expectation
For a random variable uniformly distributed over , the probability density function (PDF) is:
To calculate , we use the definition of expectation:
This is a simple integral:
Thus, the expectation is:
2. Variance
Variance is defined as:
a. First, calculate :
This integral evaluates similarly to the expectation of :
b. Now, calculate the variance:
Simplifying the expression:
Final Results:
- .
Would you like a more detailed explanation or have any questions?
Here are five related questions:
- How does the uniform distribution affect the expectation calculation?
- What changes if is uniformly distributed over instead of ?
- How would the expectation change if followed a different distribution, like exponential?
- What is the significance of calculating ?
- How does the integral method change for non-uniform distributions?
Tip: For uniform distributions, integrals involving powers of often follow simple patterns, making calculations straightforward.
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Math Problem Analysis
Mathematical Concepts
Probability
Uniform Distribution
Expectation
Variance
Formulas
E(X^n) = ∫_0^1 x^n dx = 1 / (n + 1)
Var(X^n) = E(X^2n) - (E(X^n))^2 = (1 / (2n + 1)) - (1 / (n + 1))^2
Theorems
Definition of Expectation
Definition of Variance
Suitable Grade Level
Undergraduate
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