52.2.7 Exponential Random Variable

The exponential distribution is the probablity distribution of the time between events in a process where the events occur continuously and independently at a constant average rate.

Function: pdf_exp (x,m) ¶

Returns the value at x of the density function of an \({\it Exponential}(m)\) random variable, with \(m>0\).

The \({\it Exponential}(m)\) random variable is equivalent to the \({\it Weibull}(1,1/m)\) .

The pdf is

\[f(x; m) = \cases{ me^{-mx} & for $x \ge 0$ \cr 0 & otherwise } \]

(%i1) load ("distrib")$

(%i2) pdf_exp(x,m);
                         - m x
(%o2)                m %e      unit_step(x)

Categories: Package distrib ·

Function: cdf_exp (x,m) ¶

Returns the value at x of the distribution function of an \({\it Exponential}(m)\) random variable, with \(m>0\).

The \({\it Exponential}(m)\) random variable is equivalent to the \({\it Weibull}(1,1/m)\) .

The cdf is

\[F(x; m) = \cases{ 1 - e^{-mx} & $x \ge 0$ \cr 0 & otherwise } \]

(%i1) load ("distrib")$

(%i2) cdf_exp(x,m);
                          - m x
(%o2)              (1 - %e     ) unit_step(x)

Categories: Package distrib ·

Function: quantile_exp (q,m) ¶

Returns the q-quantile of an \({\it Exponential}(m)\) random variable, with \(m>0\); in other words, this is the inverse of cdf_exp. Argument q must be an element of \([0,1]\).

The \({\it Exponential}(m)\) random variable is equivalent to the \({\it Weibull}(1,1/m)\) .

(%i1) load ("distrib")$

(%i2) quantile_exp(0.56,5);
(%o2)                  0.1641961104139661

(%i3) quantile_exp(0.56,m);
                       0.8209805520698303
(%o3)                  ------------------
                               m

Categories: Package distrib ·

Function: mean_exp (m) ¶

Returns the mean of an \({\it Exponential}(m)\) random variable, with \(m>0\).

The \({\it Exponential}(m)\) random variable is equivalent to the \({\it Weibull}(1,1/m)\) .

The mean is

\[E[X] = {1\over m} \]

(%i1) load ("distrib")$

(%i2) mean_exp(m);
                                1
(%o2)                           -
                                m

Categories: Package distrib ·

Function: var_exp (m) ¶

Returns the variance of an \({\it Exponential}(m)\) random variable, with \(m>0\).

The \({\it Exponential}(m)\) random variable is equivalent to the \({\it Weibull}(1,1/m)\) .

The variance is

\[V[X] = {1\over m^2} \]

(%i1) load ("distrib")$

(%i2) var_exp(m);
                               1
(%o2)                          --
                                2
                               m

Categories: Package distrib ·

Function: std_exp (m) ¶

Returns the standard deviation of an \({\it Exponential}(m)\) random variable, with \(m>0\).

The \({\it Exponential}(m)\) random variable is equivalent to the \({\it Weibull}(1,1/m)\) .

The standard deviation is

\[D[X] = {1\over m} \]

(%i1) load ("distrib")$

(%i2) std_exp(m);
                                1
(%o2)                           -
                                m

Categories: Package distrib ·

Function: skewness_exp (m) ¶

Returns the skewness coefficient of an \({\it Exponential}(m)\) random variable, with \(m>0\).

The \({\it Exponential}(m)\) random variable is equivalent to the \({\it Weibull}(1,1/m)\) .

The skewness coefficient is

\[SK[X] = 2 \]

(%i1) load ("distrib")$

(%i2) skewness_exp(m);
(%o2)                           2

Categories: Package distrib ·

Function: kurtosis_exp (m) ¶

Returns the kurtosis coefficient of an \({\it Exponential}(m)\) random variable, with \(m>0\).

The \({\it Exponential}(m)\) random variable is equivalent to the \({\it Weibull}(1,1/m)\) .

The kurtosis coefficient is

\[KU[X] = 6 \]

(%i1) load ("distrib")$

(%i2) kurtosis_exp(m);
(%o2)                           6

Categories: Package distrib ·

Function: random_exp (m)
    random_exp (m,k) ¶

Returns an \({\it Exponential}(m)\) random variate, with \(m>0\). Calling random_exp with a second argument k, a random sample of size k will be simulated.

The simulation algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

Categories: Package distrib · Random numbers ·