Probability density function

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

1.
The total area under the probability density curve is 1.
2.
area under the probability density curve between a and b.
3.
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,

The standarized variable

has mean 0 and sd 1.