if x is uniform over (0 1) calculate e x and var(xn)

If \( x \) is uniformly distributed over \( (0, 1) \), we denote it as \( x \sim \text{Uniform}(0, 1) \).

### 1. Expected Value \( E(x) \)

The expected value of a uniform random variable \( x \) over \( (0, 1) \) is calculated as:

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

This is a simple integration:

\[
E(x) = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2}
\]

Thus, \( E(x) = \frac{1}{2} \).

### 2. Variance of \( x^n \)

To calculate the variance of \( x^n \), we first need \( E(x^n) \) and \( E(x^{2n}) \).

#### a) \( E(x^n) \)

The expected value of \( x^n \) is:

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

Solving this integral:

\[
E(x^n) = \left[ \frac{x^{n+1}}{n+1} \right]_0^1 = \frac{1^{n+1}}{n+1} - \frac{0^{n+1}}{n+1} = \frac{1}{n+1}
\]

Thus, \( E(x^n) = \frac{1}{n+1} \).

#### b) \( E(x^{2n}) \)

Similarly, for \( x^{2n} \):

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

Thus, \( E(x^{2n}) = \frac{1}{2n+1} \).

#### c) Variance \( \text{Var}(x^n) \)

The variance of \( x^n \) is given by:

\[
\text{Var}(x^n) = E(x^{2n}) - \left( E(x^n) \right)^2
\]

Substituting the values we found:

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

Simplifying the expression:

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

### Final Result

- \( E(x) = \frac{1}{2} \)
- \( \text{Var}(x^n) = \frac{1}{2n+1} - \frac{1}{(n+1)^2} \)

Would you like further details on any part of this? 

Here are five related questions to consider:
1. How does the uniform distribution affect other moments of \( x \)?
2. What happens to \( E(x^n) \) as \( n \to \infty \)?
3. Can you derive the skewness for \( x^n \)?
4. How does the variance of \( x^n \) behave for large \( n \)?
5. What is the median of \( x \) when uniformly distributed over \( (0, 1) \)?

**Tip:** When working with uniform distributions, the expected value is always the midpoint of the interval, and higher moments follow simple polynomial patterns.

Explore the calculation of expected value E(x) and the variance of x^n for a uniform random variable over (0, 1). This includes key formulas and detailed steps for finding E(x^n) and Var(x^n). Ideal for college-level probability students.

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