Introduction
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X or just distribution function of X, evaluated at x, the probability that random variable X will have a value less than or equal to x. The cumulative distribution function is the function brother of PDF. We generally use CDFs on PDF- distributed data.
The unique aspect of CDFs is that they are monotonic. More specifically, monotonic increasing,i.e., it means that the probability will always increase over time, so CDFs are typically used as a scalar of a continuous distribution.
In the case of a continuous scalar distribution, it gives the area under the probability density function from minus infinity to x.
The CDF of a real-valued X is given by
| Fx(x) = P(X<=x) |
The right side of this equation represents the probability that random variable X takes a value less than or equal to X.
The possibility that X lies in the interval (a, b] where a < b.
| P(a<X<=b)=Fx(b) - Fx(a) |
In simpler terms, we check the probability of X being less than or equal to x. The graph of a CDF is a straight line. Otherwise, it will go up. That is why CDF has a hill-like shape, and precisely that is why CDF is monotonic non-decreasing. CDFs are always right-continuous, so the values continuously go left to right. So if we plot them, they would increase from left to right.
The CDFs of a continuous random variable X can be expressed as the integral of its probability density function fx as follows:
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In the case of a random variable X that has discrete distribution, CDF is as follows:
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