Densities 10/2

A continuous random variable takes on a non-countable infinity of possible values, here we will define it with the help of a density function.

A density is a continuous non-negative function defined on all the reals and such its integral is equal to 1.

$$P\{X\in B\}=\int_B f(x)dx$$

For B=[a,b] an interval:

$$P\{a\leq X \leq b\}=\int_a^b f(x)dx$$

If we take b=a, we see that for all a, the probability that a continuous random variable takes on that value is 0, this is the big difference with discrete random variables.

$$P\{a\leq X \leq a\}=\int_a^a f(x)dx=0$$

This implies:

$$P\{a\leq X \leq b\}=P\{a\leq X < b\}= P\{a < X < b\}$$

Intuitively for a very small width $\delta x$ the probability will be proportional to the density at x:

$$P\{x\leq X \leq x+\delta x\}\sim f(x) \delta x$$