卡方分布笔记

The chi-square distribution is a continuous probability distribution that is widely used in statistical inference, particularly in the context of hypothesis testing and in the construction of confidence intervals. It arises primarily in the context of estimating the variance of a normally distributed population and in the testing of independence in contingency tables.

Here are some key points about the chi-square distribution:

1. **Definition:** The chi-square distribution with \( k \) degrees of freedom is the distribution of a sum of the squares of \( k \) independent standard normal random variables. If \( Z_1, Z_2, ..., Z_k \) are independent standard normal random variables, then the random variable \( Q \), defined as \( Q = Z_1^2 + Z_2^2 + ... + Z_k^2 \), follows a chi-square distribution with \( k \) degrees of freedom, denoted as \( Q \sim \chi^2(k) \).

2. **Shape:** The shape of the chi-square distribution depends on the degrees of freedom. With 1 degree of freedom, the distribution is heavily skewed to the right, but as the degrees of freedom increase, the distribution becomes more symmetric and approaches a normal distribution.

3. **Non-negativity:** Since it is defined as a sum of squares, the chi-square distribution is always non-negative.

4. **Applications:**
- In hypothesis testing, it is used for tests of independence and goodness-of-fit.
- In confidence interval estimation, it helps to construct intervals for population variances.
- It's also used in the analysis of variance (ANOVA).

5. **Probability Density Function (PDF):** The PDF of a chi-square distribution with \( k \) degrees of freedom is given by:

\[ f(x; k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{(k/2)-1} e^{-x/2} \]

for \( x > 0 \), where \( \Gamma \) denotes the gamma function, which extends the factorial function to non-integer values.

6. **Mean and Variance:** The mean of a chi-square distribution is equal to its degrees of freedom (\( k \)), and its variance is twice its degrees of freedom (\( 2k \)).

The chi-square distribution is a special case of the gamma distribution and plays a crucial role in various statistical methodologies.