Download Random Variables

1.2
Random Variables
Most applications of probability theory are phrased in terms of random variables.
Definition 1. T is a random variable if T is a Borel measurable real-valued function on
. We shall abbreviate the set {  : T()  t} by {T  t} and similarly in other cases.
Definition 2. If T is a random variable, then the probability measure PT on R induced by
T is
PT(E) = P{T  E} = probability of the value of T lying in E
The distribution function of T is the function
F(t) = P{T  t} = PT( (- , t] )
The survival (reliability, complementary distribution) function of T is the function
S(t) = P{T > t} = PT( (t, ) ) = 1 - F(t).
Propostion 1. If E is a Borel set then PT(E) = 
 dF(t)
E
Proof. As E varies both and PT(E) and 
 dF(t) are measures on the Borel sets. These
E
two measures agree for sets E of the form (- , t], so the agree for all Borel sets. //
Proposition 2. The distribution function F(t) has the following properties.
(1)
F(t) is non-decreasing
(2)
F(t)  0 as t  - 
(3)
F(t)  1 as t  
(4)
F(t) is continuous from the right, i.e. F(t)  F(s) as t  s
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(5)
P{T < t} = F(t -)
The survival function S(t) has the following properties
(6)
S(t) is non-increasing
(7)
S(t)  1 as t  - 
(8)
S(t)  0 as t  
(9)
S(t) is continuous from the right, i.e. S(t)  S(s) as t  s
(10)
P{T  t} = S(t -)
Proof. Suppose s  t. Then {T  s}  {T  t}, so P{T  s}  P{T  t} which implies
F(s)  F(t) which proves (1). To prove (2), suppose tn  - . Then as n increases {T  tn}

is a decreasing sequence of sets with  {T  tn} = . So P{T  tn}  P() = 0. Thus
n=1
F(tn)  0 which proves (2). To prove (3), suppose tn  . Then as n increases {T  tn} is

an increasing sequence of sets with  {T  tn} = . So P{T  tn}  P() = 1. Thus
n=1
F(tn)  1 which proves (3). To prove (4), suppose tn  s. Then as n increases {T  tn} is a

decreasing sequence of sets with  {T  tn} = {T  s}. So P{T  tn}  P{T  s}. Thus
n=1
F(tn)  F(s) which proves (4). To prove (5), suppose tn  t. Then as n increases {T  tn}

is an increasing sequence of sets with  {T  tn} = {T < t}. So P{T  tn}  P{T < t}.
n=1
Thus F(tn)  {T < t}. However, F(tn)  F(t -). So (5) follows. The relations (6) – (10)
follow from (1) – (5) and the fact that S(t) = 1 - F(t). 
Theorem 3. Suppose a function F(t) satisfies (1) – (4). Let F be the Borel subsets of R.
Then there is a probability measure P on (R, F) such that F(t) = P{ (- , t] }. If T(t) = t,
then F(t) is the distribution function of the random variable T.
Proof. For an equivalent theorem, see Loeve [10, p. 97].
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Definition 3. The essential supremum of T, denoted by supT, is the essential supremum
of T considered as a measurable real valued function on , i.e
supT = min{t: P(T > t) = 0}
= smallest t such that the probability is zero that T > t
= min{t: S(t) = 0}
= min{t: F(t) = 1}
= smallest t such that the probability is one that T  t
Note: supT =  if P(T > t) > 0 for all t.
Definition 4. The cummulative hazard function of T is the function
(t) =



- log[ S(t) ]

if t < supT
if t  supT
= - log S(t)
with the convention that log 0 = - 
Proposition 4. With the convention that e- = 0 one has
(11)
S(t) = e-(t)
Proof. This is an immediate consequence of the definition of (t). 
Proposition 5. The cummulative hazard function (t) has the following properties.
(12)
(t) is non-decreasing
(13)
(t)  0 as t  - 
(14)
(t)   as t  
(15)
(t) is continuous from the right, i.e. (t)  (s) as t  s
Proof. These follow from (6) – (9). 
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Definition 5. If T is a random variable, then T is a continuous if and only if F(t) is
absolutely continuous. In that case the density function of T is the function
f(t) = F'(t) = - S'(t)
and the hazard (mortality, decay) rate function of T is the function h(t) defined for
t < supT by
f(t)
= S(t)
h(t) = '(t)
Proposition 6.
If T is continuous then
PT(E) = 
 f(t) dt
E
t
(16)
F(t) = 
 f(s) ds
-

(17)
 f(s) ds
S(t) = 
t
t
(18)
(t) = 
 h(s) ds
if t < supT
-
t
(19)
S(t) = exp[ - 
 h(s) ds ]
if t < supT
-
Proof. Since F(t) is absolutely continuous and f(t) is the derivative of F(t) one has
t
F(t) - F(a) = - 
 f(s) ds. Letting a  -  and using the fact that F(t)  0 as t  -  one
a
obtains (16). (17) follows from (16) and the fact that S(t) = 1 – F(t). If t < supT then S(t)
is bounded away from zero, so (t) is absolutely continuous on (- , t]. Since the
t
 h(s) ds. Letting a  -  and
derivative of (t) is h(t) one obtains (t) - (a) = - 
a
using the fact that (t)  0 as t  -  one obtains (18). (19) follows from (18) and
Proposition 4. //
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Definition 6. If T is a random variable, then the probability mass function of T is the
function
pt
= P{T = t}
= F(t) - F(t-)
= S(t) - S(t -)
T is a discrete if and only if there is a countable set E  R such that  pt = 1
tE
Proposition 7. If T is discrete then
PT(E) =
 pt
tE
In situations involving more than one random variables, subscripts indicate which
random variable a particular function is associated with, e.g. FT(t).
Example 1. T is said to be an exponential random variable with decay rate  if its
density function is given by
f(t)
=
f(t; ) =
  e - t

0
if t  0
if t < 0
It is not hard to verifty that
F(t) =
F(t; ) =
 1 - e - t

0
if t  0
if t < 0
S(t) =
S(t; ) =
 e - t

1
if t  0
if t < 0
h(t) =
h(t; ) =


0
if t  0
if t < 0
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Proposition 7.
(a)
(b)
If c is a number
FT+c(t) = FT(t-c)
ST+c(t) = ST(t-c)
fT+c(t) = fT(t-c)
hT+c(t) = hT(t-c)
If c > 0 then
FcT(t) = FT(t/c)
ScT(t) = ST(t/c)
fcT(t) = fT(t/c)
hcT(t) = hT(t/c)
Proof. (a) FT+c(t) = Pr{: T() + c  t} = Pr{: T()  t-c} = FT(t-c). Similarly for the
other relations in (a). (b) FcT(t) = Pr{: cT()  t} = Pr{: T()  t/c} = FT(t/c).
Similarly for the other relations in (b). //
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