Discrete random variable :
From the above definition, it can be seen that the number of values of the discrete random variable is limited or can be infinite (integer or natural number).
For example, tossing a coin:
rolling a dice:
All possible results of the above experiment must meet the following conditions:
If (0, 1) is used to represent the front and back of the coin, use (1, 2, 3, 4, 5, 6) Representing the points of the dice, then, these numbers should not be just numbers in the mathematical sense, but each number represents one thing: 0 represents the back of the coin, 1 represents the front of the coin, etc., which is what we call events, the probability of a single event is not equal to 0.
Continuous random variable : Continuous random variable refers to a random variable that takes any point within a certain interval on the number axis if all the possible values of the random variable X cannot be listed one by one.
It can be seen from the above definition that continuous random variables are first of all non-enumerable, that is, real numbers ; secondly, only continuous random variables have the concept of probability density. The single-point probability of a continuous random variable is 0, because each of its points is now just a point without size in the mathematical sense, unlike a number in a discrete random variable that can represent an event.
Examples of discrete random variables:
Binomial distribution :
Each value of k in the above figure represents an event.
Each value of k in the above picture Poisson distribution also represents an event, for example:
The k in the binomial distribution and Poisson distribution is limited and can be enumerated. Each value of k represents an event. an event.
Let’s look at the example of continuous random variables:
For example, uniform distribution:
As can be seen from the probability density graph in the above figure, x here is a real number and is not listable. Each x is just a number on the number axis without size. point, it cannot represent an event, and its distribution function is:
In summary, the difference between discrete and continuous new random variables is roughly:
: Each event of a discrete random variable can be represented by a number, But this number is not a point on the real number axis in the mathematical sense, but represents an event; x of the continuous random variable is a mathematical point with no size on the real number axis in the complete sense;
: continuous random Only variables have the concept of probability density, but discrete types do not.
: The single-point probability of a continuous random variable is 0, but not that of a discrete random variable, but the probability of this event occurring.
