1. Show that the
function given below is a probability density function
for any k > 0:Virtual Laboratories > Special Distributions >1 2 3 4 5 6 7 8 9 [10] 11 12 13 14 15
In this section, we will study a two-parameter family of distributions that has special importance in reliability.
1. Show that the
function given below is a probability density function
for any k > 0:
f(t) = k tk - 1 exp(-tk), t > 0.
The distribution with the density in Exercise 1 is known as the Weibull distribution distribution with shape parameter k, named in honor of Wallodi Weibull.
2. In the
random variable
experiment, select the Weibull distribution. Vary the
shape parameter and note the shape and location of the density function. Now with k =
2, run the simulation 1000 times with an update frequency of 10. Note
the apparent convergence of the empirical density to the true density.
The following exercise shows why k is called the shape parameter.
3. Graph the
density function function. In particular, show that
4. Show that the
distribution function is
F(t) = 1 - exp(-tk), t > 0.
5.
Show that the quantile function
is
F-1(p) = [-ln(1 - p)]1/k for 0 < p < 1.
6. In the quantile
applet, select the Weibull distribution. Vary the
shape parameter and note the shape and location of the density function and the
distribution function.
7.
With k = 2, find the median and the first and third quartiles. Compute
the interquartlie range.
8. Show that the
reliability function is
G(t) = exp(-tk), t > 0.
9. Show that the
failure rate function is
h(t) = k tk - 1 for t > 0.
10. Graph the
failure rate function h, and relate the graph to that of the density function f.
In particular show that
Thus, the Weibull distribution can be used to model devices with decreasing failure rate, constant failure rate, or increasing failure rate. This versatility is one reason for the wide use of the Weibull distribution in reliability.
Suppose that X has the Weibull distribution with shape parameter k. The moments of X, and hence the mean and variance of X can be expressed in terms of the gamma function.
11. Show E(Xn)
= gam(1 + n / k) for n > 0. Hint:
In the integral for E(Xn), use the substitution u =
tk. Simplify and recognize the integral as a
gamma integral.
12. Use the result
of the previous exercise to show that
13. In the
random variable
experiment, select the Weibull distribution. Vary the
shape parameter and note the size and location of the mean/standard deviation bar. Now with k
= 2, run the simulation 1000 times with an update frequency of 10.
Note the apparent convergence of the empirical moments to the true moments.
The Weibull distribution is usually generalized by the inclusion of a scale parameter b. Thus, if Z has the Weibull distribution with shape parameter k, then X = bZ has the Weibull distribution with shape parameter k and scale parameter b.
Analogues of the results given above follow easily from basic properties of the scale transformation.
14. Show
that the density function is
f(t) = (k tk - 1 / bk) exp[-(t / b)k], t > 0.
Note that when k = 1, the Weibull distribution reduces to the exponential distribution with scale parameter b. The special case k = 2, is called the Rayleigh distribution with scale parameter b, named after William Strutt, Lord Rayleigh.
Recall that the inclusion of a scale parameter does not effect the basic shape of the density; thus the results in Exercises 3 and 10 hold, with the following exception:
15.
Show that when k > 1, the mode occurs at b [(k - 1) / k]1/k.
16. In the random variable
experiment, select the Weibull distribution. Vary the
parameters and note the shape and location of the density function. Now with k =
3 and b = 2, run the simulation 1000 times with an update frequency of 10. Note
the apparent convergence of the empirical density to the true density.
17. Show that the
distribution function
F(t) = 1 - exp[-(t / b)k], t > 0.
18.
Show that the quantile function is
F-1(p) = b [-ln(1 - p)]1/k for 0 < p < 1.
19. Show that the
reliability function is
G(t) = exp[-(t / b)k], t > 0.
20. Show that the
failure rate function is
h(t) = k tk - 1 / bk.
21. Show E(Xn)
= bn gam(1 + n / k) for n > 0.
22.
Show that
23. In the random variable
experiment, select the Weibull distribution. Vary the
parameters and note the size and location of the mean/standard deviation bar. Now with k
= 3 and b = 2, run the simulation 1000 times with an update frequency of 10.
Note the apparent convergence of the empirical moments to the true moments.
24. The lifetime T
of a device (in hours) has the Weibull distribution with shape parameter k
= 1.2 and scale parameter
There is a simple one-to-one transformation between Weibull distributed variables and exponential distributed variables.
25. Show that
The following exercise is a restatement of the fact that b is a scale parameter.
26. Suppose
that X has the Weibull distribution with shape parameter k and scale
parameter b. Show that if c > 0 then cX has the Weibull
distribution with shape parameter parameter k and scale parameter bc.
27. Suppose that (X,
Y) has the standard bivariate normal distribution.
Show that the polar coordinate distance R given below has the Rayleigh distribution with scale parameter 21/2: