What is special about discrete random variables?
How is the probability mass function defined?
How do you calculate the cumulative probability distribution function from the probability mass function?
How do you calculate the probability mass function from the cumulative distribution?
What does it mean when we state that the probability mass function is normalised?
A random variable is said to be a discrete random if it cannot take all the real values in an interval on the real axis. In other words, discrete random variables are random variables that can only take particular (usually integer) values.
The probability mass function $P(X=x)$ tells one the probability that the discrete random variable $X$ is equal to $x$. You can calculate the cumulative probability distribution function from the probability mass function as follows:
$$ P(X \le x ) = \sum_{y=0}^x P(X=y) $$
Similarly one can calculate the probability mass from the cumulative probability distribution as follows:
$$ P(X=x) = P(X \le x) - P(X \le x-1) $$
As all the possible values for the random variable are mutually exclusive the probability mass function must satisfy:
$$ \sum_{x=0}^\infty P(X=x) = 1 $$
as it must be normalised. The probability mass function for discrete random variables is introduced in the following video:
https://www.youtube.com/embed/5qGlpl76t1A
<aside> 📌 SUMMARY: The probability mass function tells you the probability that a discrete random variable is equal to a particular value. The sum of all the elements of the probability mass function is equal to one.
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