The distribution of X and the central limit theorem
The distribution of X and the central limit theorem
Distribution of X

Random variables are variables linked to a random event. The distribution of X describes the possible outcomes of the random variable X, along with their associated probabilities.

The probability distribution of a variable is detailed by the probability function, which provides the probabilities of discrete outcomes, or the probability density function for continuous variables.

A discrete probability distribution or probability mass function (pmf) lists the exact probabilities of discrete outcomes, while the probability density function (pdf) provides probabilities for ranges of outcomes for continuous outcomes.

The expected value, E(X), of a random variable X is computed as the sum of the product of each outcome and its associated probability for discrete variables, or integrated over the range for continuous variables.

The variance, Var(X), measures the dispersion of the random variable from its expected value. It is calculated as the expected value of the squared deviation from the mean.

The standard deviation is the square root of the variance, providing a measure of dispersion in the same units as the random variable.
Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental theorem in probability theory and statistics which states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution, regardless of the shape of the original distribution.

This theorem has immense importance due to the ubiquity of the Normal Distribution in applied science, engineering, mathematics and natural science.

The central limit theorem explains why many distributions tend to be close to the normal distribution. The key factor is that the average of a large number of variables, irrespective of the original distribution, follows a normal distribution.

Only two critical conditions are required for the theorem to hold: The random variables must be identically distributed, and they must be independent of each other.

The standard normal distribution, also known as the zdistribution, is a special case of the normal distribution where the mean is 0 and the standard deviation is 1.

The concept of the central limit theorem is the logical foundation for many statistical procedures, including hypothesis testing and confidence intervals, which both assume normal distribution.

Practically, it allows us to make significant inferences about population parameters using sample data.