A normal distribution, also known as a Gaussian distribution, is one of the most crucial probability distributions in statistics. It is defined by its bell-shaped curve, which is symmetric around the mean. In mathematical terms, a random variable following a normal distribution can be expressed as:
- Mean (\( ext{mean} = heta \) for our problem)
- Variance (\( ext{variance} = heta \) since the variance is tied to \( heta \) in this context)
The mean indicates the center or peak of the distribution, while the variance measures the spread or dispersion of data points around the mean.
In our exercise, each sample \( X_i \) is drawn from a normal distribution with a specified mean (\( 0 \)) and variance (\( heta \)). This implies that each data set of \( X_i^2 \) contributes to both estimating the mean \( 0 \) and the unknown variance \( heta \). The property of the normal distribution that supplements this analysis is how these sample moments relate to the theoretical moments.