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

Given that \( X \) is uniformly distributed over \( (0,1) \), we can calculate the expected value \( E(X^n) \) and variance \( \text{Var}(X^n) \).

### 1. **Expected Value \( E(X^n) \)**

For a uniform random variable \( X \sim U(0,1) \), its probability density function (PDF) is:

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

The expected value \( E(X^n) \) is defined as:

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

Since \( f_X(x) = 1 \), the integral simplifies to:

\[
E(X^n) = \int_0^1 x^n \, dx = \left[ \frac{x^{n+1}}{n+1} \right]_0^1 = \frac{1}{n+1}
\]

Thus:

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

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

The variance \( \text{Var}(X^n) \) is defined as:

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

- From the first part, we know \( E(X^n) = \frac{1}{n+1} \).
- Next, we need to compute \( E(X^{2n}) \), which is:

\[
E(X^{2n}) = \int_0^1 x^{2n} \, dx = \left[ \frac{x^{2n+1}}{2n+1} \right]_0^1 = \frac{1}{2n+1}
\]

Now, we can compute the variance:

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

Simplifying:

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

To combine these terms, find a common denominator:

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

Simplify the numerator:

\[
(n+1)^2 - (2n+1) = n^2 + 2n + 1 - 2n - 1 = n^2
\]

Thus:

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

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

Would you like further clarification or a detailed explanation on a specific step?

Here are 5 related questions to consider:
1. How does the uniform distribution impact the expected value formula for different functions of \( X \)?
2. How would you compute \( E(X^n) \) for a different distribution, such as exponential?
3. What happens to \( \text{Var}(X^n) \) as \( n \to \infty \)?
4. How would you use integration by parts to solve similar expected value problems?
5. How does the variance change if \( X \) is uniformly distributed over a different interval, say \( (a,b) \)?

**Tip**: When dealing with expectations involving powers of random variables, leveraging known integral results can save time in calculations.

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

Explore how to calculate the expected value E(X^n) and variance Var(X^n) for a random variable X uniformly distributed over (0,1). This problem involves key concepts in probability theory, including integrals of polynomials, uniform distribution, and variance computations, suitable for advanced undergraduate students.

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