What is special about continuous random variables?
How is the probability density function defined?
How do you calculate the cumulative probability distribution function from the probability density function?
What does it mean when we state that the probability density function is normalised?
A random variable is said to be a continuous random if it can take any real value in an interval on the real axis.
The probability density function $f(x)$ tells for the continuous random variable $X$ is equal to the first derivative of the cumulative cumulative probability distribution function for the variable as shown below:
$$ f(x) = \frac{\textrm{d}P(X\le x)}{\textrm{d}x} $$
One can, therefore, calculate the cumulative probability distribution function from the probability density as follows:
$$ P(X\le x) = \int_{-\infty}^x f(y) \textrm{d}y $$
As all the possible values that the random variable can take are mutually exclusive the probability density function must satisfy:
$$ \int_{-\infty}^\infty f(x) \textrm{d}x = 1 $$
as it must be normalised. The probability density function for continuous random variables is introduced in the following video:
https://www.youtube.com/embed/5qGlpl76t1A
<aside> 📌 SUMMARY: The probability density function is equal the first derivative of the cumulative probability distribution function of a continuous random variable. The integral of the probability density function over all space is equal to one.
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