Probability density function

The probability density function f(x) describes the distribution of probability for a continuous random variable. It has the properties

The total area under the probability density curve is 1.
$P[a \leq X \leq b]=$ area under the probability density curve between a and b.
$f(x) \geq 0$ for all x.

With a continuous random variable, the probability that X=x is always 0. It is only meaningful to speak about the probability that X lies in an interval P[X=x]=0, if X is a continuous random variable.

When determining the probability of an interval a to b, we need not be concerned if either or both end points are included in the interval. Since the probabilities of X=a and X=b are both equal to 0,

$P[a\leq X\leq b]=P[a<X\leq b]=P[a\leq X<b]=P[a<X<b]$
$P[a<X<b]=(\mbox{Area to left of }b)-(\mbox{Area to left of }a)$
$P[b<X]=1-(\mbox{Area to left of }b)$

The standarized variable

has mean 0 and sd 1.