The normal distribution has several characteristics that make it the underlying foundation of much of inferential statistical tools. List these characteristics and explain how they contribute to the power of the distribution as foundation of inferential decision making.
In probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that cluster around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known as the Gaussian function or bell curve.
Characteristics of Normal Distribution
1. The family of normal distributions is closed under linear transformations. That is, if X is normally distributed with mean μ and variance σ2, then a linear transform aX + b (for some real numbers a ≠ 0 and b) is also normally distributed.
2. Also if X1, X2 are two independent normal random variables, with means μ1, μ2 and standard deviations σ1, σ2, then their linear combination will also be normally distributed.
3. The converse of (1) is also true: if X1 and X2 are independent and their sum X1 + X2 is distributed normally, then both X1 and X2 ...
Why is the normal distribution important and why is it so commonly used?