The cumulative distribution function, (c.d.f)

Understanding the Cumulative Distribution Function (c.d.f)

The Cumulative Distribution Function (c.d.f) is a function that gives the probability that a random variable is less than or equal to a certain value.
The c.d.f describes the distribution of probability over the entire range of a random variable.
A c.d.f is always non-decreasing and right-continuous, which means it never decreases as the variable increases and has no jumps to the right.

The c.d.f ranges from 0 to 1. The minimum value (0) tells us that the probability of the random variable taking a value less than the smallest possible value is 0.
Similarly, the maximum value (1) tells us that the probability of the random variable taking a value less than or equal to the largest possible value is 1.
The c.d.f for a specific value can be calculated by summing up the probabilities of all values up to that point.
For discrete random variables, the c.d.f jumps by the probability of that value each time.
For continuous random variables, the c.d.f is a smooth, unbroken function.

The c.d.f of a random variable X for a specific value x can be calculated using the formula F(x) = P(X ≤ x).
For discrete random variables, the c.d.f at a specific value x can be calculated by adding up all the probabilities of the outcomes that are less than or equal to x.
For continuous random variables, the c.d.f is found by integrating the probability density function up to the value x.

The c.d.f can be used to calculate percentiles, which are the values below which a certain percent of the data falls.
It can also be used to find probabilities of intervals, by subtracting the c.d.f values at the endpoints of the interval.
The c.d.f gives a complete description of the statistical distribution of a random variable, and can be used to calculate any statistic - such as the mean, variance or median. However, this requires more advanced mathematical techniques.