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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.

\begin{displaymath}P\{X\in B\}=\int_B f(x)dx\end{displaymath}

For B=[a,b] an interval:

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

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.

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

This implies:

\begin{displaymath}P\{a\leq X \leq b\}=P\{a\leq X < b\}=
P\{a < X < b\}\end{displaymath}

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

\begin{displaymath}P\{x\leq X \leq x+\delta x\}\sim f(x) \delta x\end{displaymath}


Susan Holmes