Area under the curve - probability

[michonamona]

It's something that I know but I've never been able to figure out 'why?' Why is does the area underneath the normal distribution (or any distribution) represent probability?

Thank you.

M

 

[LCKurtz]

It's something that I know but I've never been able to figure out 'why?' Why is does the area underneath the normal distribution (or any distribution) represent probability?

Thank you.

M

The short answer to your question is the definition:

A nonnegative function f(x) is said to be the density function for the continuous random variable X if for all real numbers a < b:

$$P(a\le X \le b) = \int_a^b f(x)\, dx$$

From calculus we know the integral on the right represents the area under f(x) between a and b. Since the cumulative distribution function is given by

$$P(X \le x) =\int_{-\infty}^{x} f(t)\, dt$$

it follows that

$$\int_{-\infty}^{\infty} f(x)\, dx = 1$$

Putting these together shows why the area under the curve represents probability. Does that explanation help or did you already know that?