continuous random formula

Posted 2021-02-07 phphelper

2. The hazard or failure rate function of a non-negative continuous random variable X is
defined to be
where f(x) is the pdf of X and F(x) is its cdf. We can also define h(x) by
h(x) = lim#0
P(x X x + |Xx).
(a) A useful formula for the expected value of any non-negative random variable is
E(X) = Z 1
F(x)) dx.
If X is also continuous with pdf f(x) then this formula can be derived as follows:
E(X) = Z 1
xf(x) dx
If h(x) is the hazard function of X, show that
E(X) = Z 1
(Hint: Make the change of variables u = F 1().)
(b) Suppose that X(k) is the k-th order statistic where kn (for some 2 (0,1)) and define
Dk = X(k)
X(k1).
From lecture, we know that the distribution of n Dk is approximately
Exponential with mean 1/f(F 1()). Use this fact to show that the distribution of (nk +1)Dk is approximately Exponential with mean 1/h(F 1()). (Hint: Note that h(F 1()) =f(F 1())/(1 ).)
(c) The shape of h(x) provides useful information about the distribution not readily obvious
from the pdf and cdf; for example, if X represents the lifetime of some (say) electronic
component then a decreasing hazard function would indicate that the component improves
with age.
The total time on test (TTT) plot provides one to assess the rough shape of h(x) based
on a sample x1, ··· , xn. To construct this plot, we define
d1 = nx(1)
dk = (n
k + 1)(x(k)
x(k1))
for k = 2, ··· , n
and plot (d1 + ··· + dk)/(x1 + ··· + xn) versus k/n for k = 1, ··· , n. Using the result from
part (b), we might argue that (d1 + ··· + dk)/(x1 + ··· + xn) is an estimate of If the underlying hazard function h(x) is decreasing then the shape of these
points will be roughly convex (and lie below the 45o line) while if h(x) is increasing then the
shape of the points will be roughly concave (and lie above the 45o line).
Given data in a vector x, the TTT plot can be constructed as follows:
> x <- sort(x) # order elements from smallest to largest
> n <- length(x) # find length of x
> d <- c(n:1)*c(x[1],diff(x))
> plot(c(1:n)/n, cumsum(d)/sum(x), xlab="t", ylab="TTT")
> abline(0,1) # add 45 degree line to plot
Data on the lifetimes (in hours) of Kevlar 373/epoxy strands (subjected to constant pressure
at 90% stress level) are contained in the file kevlar.txt. Construct a TTT plot for these
data. Does the hazard function appear to be increasing or decreasing with time?
Supplemental problems (not to be handed in):
3. (a) Suppose that X has a Gamma distribution with shape parameter and scale parameter;
the density of X is
Find expressions for the skewness and kurtosis of X in terms of and .
(Do these depend on )
What happens to the skewness and kurtosis as
(b) Suppose that X1, ··· , Xn are independent and define Sn = X1 +···+Xn. Assuming that
E(X3i ) is well-defined for all i, show that the skewness of Sn is given by
skew(Sn) = Xn
i = Var(Xi). (Hint: Follow the proof given for the kurtosis identity assuming for
simplificity that E(Xi) = 0; this is more simple since E(Sn) involves a triple summation,
most of whose terms are 0.)
4. Suppose that X1, ··· , Xn are independent random variables with distribution function F
where μ = E(Xi) and 2
= Var(Xi). For some families of distributions, the variance is a

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