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

Given that \( X \) is uniformly distributed over \( (0, 1) \), we know that the probability density function (PDF) of \( X \) is:

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

### 1. **Calculate \( E(X^n) \):**

The expected value \( E(X^n) \) is given by the following integral:

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

Evaluating this integral:

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

So, the expected value is:

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

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

The variance of \( X^n \) is calculated using the formula:

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

First, calculate \( E(X^{2n}) \):

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

Now, using the previously calculated \( E(X^n) \):

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

Simplifying this expression:

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

Thus, the variance is:

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

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

Would you like further details or explanations on any part of the process?

Here are 5 related questions you might find helpful:
1. How does the uniform distribution affect expected values?
2. Can you generalize this method for other distributions?
3. What is the relationship between \( E(X^n) \) and \( E(X^{2n}) \) for uniform distributions?
4. How does the variance change as \( n \) increases?
5. What is the role of the PDF in calculating expected values?

**Tip:** When dealing with uniform distributions, the calculations often simplify because the PDF is constant over the given interval.

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

Learn how to calculate the expected value and variance of X^n when X is uniformly distributed over (0,1). This problem covers key probability concepts, including uniform distribution, expected value, and variance. Suitable for undergraduate students.

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