Continuous Random Variable

Definition We say a real function is absolutely continuous if there exists real function such that

Definition We say a random variable is continuous if its c.d.f is absolutely continuous

Definition The function such that is called the probability density function (p.d.f) of .

Now let's take a look at the probability mass function of .

For any number , let be the half open interval where the last equality comes from the continuity of probability function on monotonic sequence of sets. Now

If is continuous r.v, then because is continuous.

Therefore, p.m.f of a continuous r.v is meaningless, because it's p.m.f is 0 every where. So p.d.f is what we need to capture the spirit of , even though is . \textbf{Remark}

  1. When something has a probability 0, it doesn't mean it won't happen! As we just see, any p.m.f of a contunuous random variable is 0.

  2. Given a c.d.f , its corresponding p.d.f , is not unique. However, it rarely matters because the results we need generally come from imtergrating , and even if there are differen 's, the result after intergrating is always the same. Why? To answer this you will have to take some courses in measure theory.

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