The question in the image involves deriving properties of the k-th order statistic from a set of independent and identically distributed (iid) random variables, $X_1, X_2, \dots, X_n$, which follow a Uniform distribution on the interval $[0,1]$. Let's break down each part:

a) Derive the density of the k-th order statistic, $X_{(k)}$.

The density of the $k$-th order statistic, denoted by $f_{X_{(k)}}(x)$, for iid random variables $X_1, X_2, \dots, X_n$ uniformly distributed on $[0,1]$, is given by:

$f_{X_{(k)}}(x) = \frac{n!}{(k-1)!(n-k)!} \cdot x^{k-1} \cdot (1-x)^{n-k}, \quad 0 \leq x \leq 1.$

This formula arises from the fact that we want exactly $k-1$ values of the random sample to be smaller than $x$, one value to be equal to $x$, and the remaining $n-k$ values to be greater than $x$. The binomial coefficient accounts for the different ways to assign the ranks, and the powers of $x$ and $1-x$ represent the probabilities of being below or above $x$, respectively.

b) Identify the distribution in part a) and give its rationale and associated parameters.

The distribution in part a) follows a Beta distribution. Specifically, the k-th order statistic $X_{(k)}$ has a Beta distribution with parameters $k$ and $n-k+1$. That is:

$X_{(k)} \sim \text{Beta}(k, n-k+1).$

The rationale behind this is that the form of the density function for order statistics, after simplification, matches the probability density function of the Beta distribution, which is:

$f(x) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1}, \quad 0 \leq x \leq 1.$

Here, $\alpha = k$ and $\beta = n-k+1$.

c) Derive the normalizing constant of the distribution in part a).

For the Beta distribution $\text{Beta}(k, n-k+1)$, the normalizing constant is given by:

$\frac{n!}{(k-1)!(n-k)!}.$

This constant ensures that the total probability integrates to 1 over the interval $[0, 1]$. In this case, it matches the binomial coefficient that appears in the order statistic density.

d) Find the mean, second moment, and variance for $X_{(k)}$.

For the Beta distribution $\text{Beta}(k, n-k+1)$, the mean, second moment, and variance can be found using the properties of the Beta distribution:

1. Mean: $\mathbb{E}[X_{(k)}] = \frac{k}{n+1}.$

2. Second moment: $\mathbb{E}[X_{(k)}^2] = \frac{k(k+1)}{(n+1)(n+2)}.$

3. Variance: The variance is given by: [ \text{Var}(X_{(k)}) = \mathbb{E}[X_{(k)}^2] - (\mathbb{E}[X_{(k)}])^2 = \frac{k(n-k+1)}{(n+1)^2(n+2)}. ]

This completes the four parts of the question.

Would you like further details on any of these steps, or have any specific questions? Here are five related questions for deeper understanding:

1. How would you generalize this result to other distributions, not just Uniform?
2. Can you explain how the form of the Beta distribution appears in other applications of statistics?
3. What is the intuition behind the binomial coefficient in the density of the order statistic?
4. How do the moments of the Beta distribution vary as $n$ grows larger?
5. How can you apply order statistics to real-world problems, such as in statistical inference?

Tip: For understanding order statistics, it's helpful to start with small sample sizes (e.g., $n = 3$) and compute specific examples manually, then generalize to larger $n$.