https://r2.erweima.ai/mathbot/image/1727008306452b5nwuaiv.jpg

The problem provided is in Chinese and involves a probability density function and sample distribution. Here's a breakdown of the problem based on the image:

**Problem:**
- The population \( X \) has a probability density function \( f(x) \):
  \[
  f(x) =
  \begin{cases}
  2x, & 0 < x \leq 1, \\
  0, & \text{otherwise}.
  \end{cases}
  \]
- \( X_1, X_2, \dots, X_n \) are independent and identically distributed (i.i.d.) random samples from this population.
- \( F_n(x) \) is the empirical distribution function based on the sample.
- The task is to find the variance of \( F_n\left(\frac{1}{2}\right) \), denoted as \( D\left(F_n\left(\frac{1}{2}\right)\right) \).

### Steps to Solve:
1. **Distribution Overview:**
   - The given probability density function \( f(x) = 2x \) over \( (0, 1] \) is a triangular distribution.
   - The cumulative distribution function (CDF) \( F(x) \) can be found by integrating \( f(x) \):
     \[
     F(x) = \int_0^x 2t \, dt = x^2, \quad 0 < x \leq 1.
     \]

2. **Calculate \( F_n\left(\frac{1}{2}\right) \):**
   - For \( x = \frac{1}{2} \), the CDF is:
     \[
     F\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}.
     \]

3. **Variance of \( F_n(x) \):**
   - \( F_n(x) \) is the empirical distribution function, which for i.i.d. samples follows a binomial distribution:
     \[
     F_n(x) \sim \text{Binomial}\left(n, F(x)\right).
     \]
   - The variance of the empirical CDF at a specific point \( x = \frac{1}{2} \) is given by:
     \[
     \text{Var}\left(F_n\left(\frac{1}{2}\right)\right) = \frac{F\left(\frac{1}{2}\right)\left(1 - F\left(\frac{1}{2}\right)\right)}{n}.
     \]
   - Substituting \( F\left(\frac{1}{2}\right) = \frac{1}{4} \):
     \[
     \text{Var}\left(F_n\left(\frac{1}{2}\right)\right) = \frac{\frac{1}{4} \cdot \left(1 - \frac{1}{4}\right)}{n} = \frac{\frac{1}{4} \cdot \frac{3}{4}}{n} = \frac{3}{16n}.
     \]

Thus, the variance \( D\left(F_n\left(\frac{1}{2}\right)\right) = \frac{3}{16n} \).

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Let me know if you need further clarification or additional steps!

Here are 5 related questions to deepen your understanding:
1. How is the cumulative distribution function (CDF) derived from the probability density function (PDF)?
2. What is the role of the empirical distribution function in statistics?
3. Can you explain how the variance of the empirical CDF is calculated for any value of \( x \)?
4. What is the significance of using i.i.d. random samples in this context?
5. How would the variance change if the sample size \( n \) increases?

**Tip:** The variance of the empirical CDF decreases as the sample size \( n \) increases, leading to more precise estimates of the true CDF.

设总体X的概率密度函数为f(x) = {2x, 0 < x <= 1; 0, 其他}, X_1, X_2, ..., X_n是来自该总体的样本, F_n(x)为样本的经验分布函数. 试求F_n(1/2)的方差D(F_n(1/2)).

This problem involves calculating the variance of the empirical distribution function F_n at x = 1/2 for a triangular distribution with PDF f(x) = 2x on the interval (0, 1]. The solution involves concepts such as probability density functions, cumulative distribution functions, and the binomial distribution.

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