Description
1. Suppose a discrete random variable T taking values 1, 3, 5, 7, 9, 12 with probabilities 1
6
,
1
3
,
1
4
,
1
8
,
1
16 and 1
16 , respectively.
(a) Find the mean of T
(b) Find the survival function of T
(c) Find the area under the curve S(t) in the right upper quadrant, i.e., find
Z ∞
0
S(t)dt
(d) Compare results in (a) and (c).
2. Suppose the lifetime of a electronic component is exponentially distributed with
rate θ with density function
f(t) = θ exp(−θt), t > 0
Find the conditional probability that T > t + s given T ≥ t, where s > 0. Also
find the probability that T > s.
3. A random variable T is said to be Weibull distributed if its hazard function is
of the form
h(t) = αλtα−1
, t > 0
where α and λ are positive constants. Find the distribution of Y = log T.
4. Assume the lifetime random variable T is continuous and denote its mean of
remaining lifetime given T ≥ t by m(t). Then show that the following functions
of T can be expressed in term of m(t)
(a) The survival function
S(t) = m(0)
m(t)
exp ”
−
Z t
0
du
m(u)
#
(b) The density function
f(t) = (m0
(t) + 1)
m(0)
m(t)
2
!
exp ”
−
Z t
0
du
m(u)
#
(c) The hazard function
h(t) = −
d
dt log [S(t)] = m0
(t) + 1
m(t)
5. For the geometric random variable with probability mass function
Pr(X = j) = (1 − p)
j−1
p, j = 1, 2, . . . ,
find its hazard function.
6. For the Poisson distribution with probability mass function
Pr(X = j) = e
−λ λ
j
j!
, j = 0, 1, 2, . . . .
Show that the hazard function is monotone increasing.
7. Suppose that the mean residual life of a continuous survival time T is given by
m(t) = t + 10.
(a) Find the mean of T
(b) Find h(t)
(c) Find S(t)
8. Find the survival function of the Gompertz random variable where its hazard
function is given by
h(t) = θeαt, t ≥ 0; θ, α > 0.
