If X is uniform over (0,1), calculate E(Xn) and Var(Xn)

Given that \( X \) is uniformly distributed over \( (0, 1) \), and we want to calculate \( E(X^n) \) and \( \text{Var}(X^n) \), we can proceed as follows:

### 1. **Expectation \( E(X^n) \)**

For a random variable \( X \) uniformly distributed over \( (0, 1) \), the probability density function (PDF) is:

\[
f_X(x) = 1 \quad \text{for} \quad 0 \leq x \leq 1.
\]

To calculate \( E(X^n) \), we use the definition of expectation:

\[
E(X^n) = \int_0^1 x^n f_X(x) \, dx = \int_0^1 x^n \, dx.
\]

This is a simple integral:

\[
E(X^n) = \left[ \frac{x^{n+1}}{n+1} \right]_0^1 = \frac{1}{n+1}.
\]

Thus, the expectation is:

\[
E(X^n) = \frac{1}{n+1}.
\]

### 2. **Variance \( \text{Var}(X^n) \)**

Variance is defined as:

\[
\text{Var}(X^n) = E((X^n)^2) - (E(X^n))^2.
\]

#### a. **First, calculate \( E((X^n)^2) \):**

\[
E((X^n)^2) = E(X^{2n}) = \int_0^1 x^{2n} \, dx.
\]

This integral evaluates similarly to the expectation of \( X^n \):

\[
E(X^{2n}) = \frac{1}{2n+1}.
\]

#### b. **Now, calculate the variance:**

\[
\text{Var}(X^n) = \frac{1}{2n+1} - \left( \frac{1}{n+1} \right)^2.
\]

Simplifying the expression:

\[
\text{Var}(X^n) = \frac{1}{2n+1} - \frac{1}{(n+1)^2}.
\]

### Final Results:
- \( E(X^n) = \frac{1}{n+1} \)
- \( \text{Var}(X^n) = \frac{1}{2n+1} - \frac{1}{(n+1)^2} \).

Would you like a more detailed explanation or have any questions?

Here are five related questions:
1. How does the uniform distribution affect the expectation calculation?
2. What changes if \( X \) is uniformly distributed over \( (a, b) \) instead of \( (0, 1) \)?
3. How would the expectation change if \( X \) followed a different distribution, like exponential?
4. What is the significance of calculating \( \text{Var}(X^n) \)?
5. How does the integral method change for non-uniform distributions?

**Tip**: For uniform distributions, integrals involving powers of \( X \) often follow simple patterns, making calculations straightforward.

If X is uniform over (0,1), calculate E(X^n) and Var(X^n).

This page provides a step-by-step solution to calculating the expectation E(X^n) and variance Var(X^n) for a uniformly distributed random variable X over (0,1). Key concepts covered include uniform distribution, integration techniques, and formulas for expectation and variance. Suitable for undergraduate-level students.

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