Given a positive constant , the exponential density
function (with parameter
) is
Let be a continuous random variable with
an exponential density function with parameter
.
Integrating by parts with and
so that
and
, we find
Integrating by parts with and
so that
and
, we have
So,
.
Exponential random variables (sometimes) give good models for the time to failure of
mechanical devices. For example, we might measure the number of miles traveled by a
given car before its transmission ceases to function. Suppose that this distribution
is governed by the exponential distribution with mean . What is the probability
that a car's transmission will fail during its first
miles of operation?
The normal density function with mean and standard deviation
is
As suggested, if has this density, then
and
.
The standard normal density function is the normal density function with
. That is,
Let
be the
standard normal density function and let
be the standard normal cumulative distribution function.
We compute a Taylor series expansion,
So
for some
.
As
is the expected value, we need
.
If the continuous random variable is normally distributed, what is the probability that
it takes on a value of more than a standard deviations above the mean?
Via a change of variables, we may suppose that is normally distributed
with respect to the standard normal distribution. Let
be the
cumulative distribution function for the standard normal distribution.