The Chi-square distribution is an important continuous probability distribution that is widely used in statistical analysis. It is a distribution of the sum of squares of a set of independent random variables that follow a standard normal distribution.

Properties of Chi-square distribution:
1. The Chi-square distribution is always non-negative.
2. It is a right-skewed distribution with a single mode.
3. The mean and variance of the Chi-square distribution are equal to the degrees of freedom (df).
4. The shape of the distribution changes with the degrees of freedom.
5. It is used to test the goodness of fit of a sample to a theoretical distribution, test the independence of two categorical variables, and test the difference between two variances.

Applications of Chi-square distribution:
1. In the field of genetics, the Chi-square test is used to determine if the observed distribution of genotypes or phenotypes in a population fits an expected distribution.
2. In quality control, the Chi-square test is used to determine if a set of observed data follows a specific distribution.
3. In finance, the Chi-square distribution is used to model the distribution of stock prices and interest rates.

Conclusion:
In conclusion, the Chi-square distribution is an important continuous probability distribution that has a wide range of applications in various fields. It is used to test the goodness of fit, independence, and difference between two variances. Understanding the properties and applications of the Chi-square distribution is essential for statistical analysis.