Expected mean E(X), the variance Var(X), cumulative distribution function F(X)

Understanding Expected Mean (E(X))

Expected Mean (E(X)) is the weighted average of a probability distribution. It gives you the average outcome you would expect if an experiment were to be repeated multiple times.
Simply put, it’s the most probable outcome from a statistical experiment.
In the context of a discrete random variable, this can be calculated as the sum of all values of the variable each multiplied by their respective probability.
For a continuous random variable, the expected value is determined by integrating the product of the variable with its probability density function across the variable’s range.

Variance (Var(X)) aims to quantify the spread or dispersion of a set of data points around their mean value.
A large variance indicates that the data points are very spread out from the mean, and from each other. In contrast, a small variance indicates the opposite.
For a discrete random variable, the variance is calculated by summing the squared deviation of each possible outcome from the expected value, each multiplied by its corresponding probability.
For a continuous random variable, variance is determined by integrating the squared deviation of the variable from its expected value, multiplied by its probability density function across the variable’s range.

The Cumulative Distribution Function (F(X)) for a random variable is defined as the probability that the variable takes a value less than or equal to a certain value.
It summarises the probability distribution of a random variable in one function.
This function is always non-decreasing and right-continuous, which means it never decreases in value and has no jumps.
For discrete random variables, the cumulative distribution function is defined as the sum of probabilities of all outcomes less than or equal to the given value.
In the case of continuous random variables, it’s calculated by integrating the probability density function from the lowest possible value of the random variable up to the given value.
The cumulative distribution function of any random variable always ranges between 0 and 1.