Probability is a measure of the certainty in which an event might occur. This definition is easily implemented when dealing with several distinct events. When a continues random variable is examined, however, it becomes harder to use this definition directly, because the term "event" is somewhat elusive.
If we want to define the probability of an event in which the random variable takes a specific value, the probability for this event will usually be zero, because there are infinitely many distinct values in any continues interval. And it is not very useful to define probability as having a value of zero everywhere.
Instead, we define the probability of events in which the random variable takes a value within a specific interval. The customary way to do it is by using a cumulative distribution function (CDF), which is defined as the probability that the random variable takes a value smaller than some specific value xx. Mathematically we define F(x)=P(X<x)F(x)=P(X
CDF is the way the probability of a continues random variable is defined. The problem with it is that it is hard to use. We can use it to calculate the probability of the random variable taking a value within any specific range, but when we are looking for additional properties of the r.v., like mean and variance, the CDF is not enough. The CDF only gives probabilities for compound events, and in order to calculate many properties, we need some measure of the probability of the most distinctive events we can define. But we've seen before, that this probability is zero.
We know that for discrete r.v.s, the probability of the occurrence of one of several distinct events is the sum of their probabilities.
The equivalent of sum in the continues world is the integral. Therefore, we can assume that the CDF might be the integral of some function that would be useful for us, so we define
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