1. Use the change of variables theorem to show that the lognormal density
function, with parameters µ and d, is given by Virtual Laboratories > Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 [14] 15
A random variable X is said to have the lognormal distribution, with parameters µ and d, if ln(X) has the normal distribution with mean µ and standard deviation d. Equivalently,
X = exp(Y)
where Y is normally distributed with mean µ and standard deviation d. Recall that the parameter µ can be any real number while the parameter d must be a positive real number. The lognormal distribution is used to model continuous random quantities when the distribution is believed to be skewed, such as certain income and lifetime variables.
1. Use the change of variables theorem to show that the lognormal density
function, with parameters µ and d, is given by
f(x) = exp{-[ln(x) - µ]2 / (2d2)]
/ [x (2
)1/2
d] for x > 0.
2. Show that the lognormal distribution is unimodal and skewed right.
Specifically, show that
0 as x
.
0 as x
0+.
3. In the random variable
experiment, select the lognormal distribution. Vary the
parameters and note the shape and location of the density function. Now with µ
= 0 and d = 1, run the simulation 1000 times with an update frequency of 10. Note
the apparent convergence of the empirical density to the true density.
Let G denote the standard normal distribution function. Recall that values of G are tabulated and can be obtained from the quantile applet. Thus, the following exercises show how to compute the lognormal distribution function and quantiles in terms of the standard normal distribution function and quantiles.
4. Show that the lognormal distribution function F is given by
F(x) = G{[-µ + ln(x)] / d} for x > 0.
5. Show that the quantile function is given by
F-1(p) = exp[µ + d G-1(p)] for 0 < p < 1.
6. Suppose that the income X of a randomly chosen person in a
certain population (in $1000 units) has the lognormal distribution with
parameters µ = 2 and d = 1. Find P(X > 20).
7. In the quantile
applet, select the lognormal distribution. Vary the
parameters and note the shape and location of the density function and the
distribution function. With µ = 0 and d = 1, find the median and the
first and third quartiles.
The moments of the lognormal distribution can be computed from the moment generating function of the normal distribution.
8. Suppose that X has the lognormal distribution with parameters
µ and d. Show that
E(Xn) = exp(nµ + n2d2 / 2).
9. In particular, show that mean and
variance of X are
Even though the lognormal distribution has finite moments of all orders, the moment generating function is infinite at any positive number. This property is one of the reasons for the fame of the lognormal distribution.
10. Show that E[exp(tX)] =
for any t > 0.
11. Suppose that the income X of a randomly chosen person in a
certain population (in $1000 units) has the lognormal distribution with
parameters µ = 2 and d = 1. Find
12. In the
simulation of the random variable
experiment, select the lognormal distribution. Vary the
parameters and note the shape and location of the mean/standard deviation bar. Now with
µ = 0 and d = 1, run the simulation 1000 times with an update frequency of 10. Note
the apparent convergence of the empirical moments to the true moments.
The most important transformations are the ones in the definition: if X has a lognormal distribution then ln(X) has a normal distribution; conversely if Y has a normal distribution then exp(Y) has a lognormal distribution.
13. For fixed d, show that the lognormal distribution with parameters µ
and d
is a scale family with scale parameter exp(µ).
14. Show that the lognormal distribution is a 2-parameter exponential
family with natural parameters and natural statistics given by