A normal distribution is a probability distribution that is symmetrical and bell-shaped, with the majority of values falling near the mean and fewer values further away from the mean. It is also known as a Gaussian distribution or a bell curve.
To prove that a variable X follows a normal distribution, we need to show that its probability density function (PDF) follows the equation for the normal distribution: ф(x) = (1/σ√2π) * e^(-(x-m)^2/2σ^2), where m is the mean and σ is the standard deviation of X.
This equation represents the standardization of a normal distribution, where X is a random variable, m is the mean of X, and σ is the standard deviation of X. This standardization transforms any normal distribution into a standard normal distribution with a mean of 0 and a standard deviation of 1.
A normal distribution can be visually represented using a histogram or a bell-shaped curve. The x-axis represents the values of the variable, and the y-axis represents the frequency or probability of those values occurring. The curve will be symmetrical and centered around the mean, with the tails getting closer to the x-axis as they move away from the mean.
The normal distribution is significant in statistics as it is used to model many natural phenomena, such as heights, weights, and test scores. It is also used as a basis for many statistical tests and methods, making it an important tool for analyzing and interpreting data.