A random variable X is said to have Erlang-truncated exponential distribution if its probability density function (pdf) is of the form

and the corresponding distribution function is

where

.

For more details on this distribution and its applications one may refer to [1].

[2] introduced and extensively studied the generalized order statistics. The order statistics, sequential order statistics, Stigler’s order statistics, record values are special cases of. Suppose are n from an absolutely continuous distribution function with the corresponding probability density function . Their joint pdf is

for,

and is a positive integer.

Choosing the parameters appropriately, models such as ordinary order statistics (), k-th record values

, sequential order statistics, order statistics with non-integral sample size

, Pfeifer’s record values

and progressive type II censored order statistics are obtained [2, 3].

The marginal of the r-th, , , is

and the joint of and, , is

where

and.

[4-6] have established recurrence relations for moment generating functions of record values from Pareto and Gumble, power function and extreme value distributions.

Recurrence relations for marginal and joint moment generating functions of from power function distribution are derived by [7]. [8,9] have established recurrence relations for conditional and joint moment generating functions of based on mixed population, respectively. [10] has established explicit expressions and some recurrence relations for moment generating function of gos from Gompertz distribution.

In the present study, we establish exact expressions and some recurrence relations for marginal and joint moment generating functions of gos from Erlang-truncated exponential distribution. Results for order statistics and record values are deduced as special cases and a characterization of this distribution is obtained by using the conditional expectation of function of gos.

Note that for Erlang-truncated exponential distribution defined in (1).

The relation in (6) will be exploited in this paper to derive exact expressions and some recurrence relations for the moment generating functions of from the Erlang-truncated exponential distribution.

Let us denote the marginal moment generating functions of by and its j-th derivative by.

We shall first establish the explicit expression for. Using (4) and (6), we have when

where

On expanding

binomially in (8), we get when

where

On substituting for from (2) in (9), we have

Now on substituting for from (10) in (7) and simplifying, we obtain when

When, we have

Since (11) is of the form at, therefore, we have

Differentiating numerator and denominator of (12) times with respect to, we get

On applying L’ Hospital rule, we have

But for all integers and for all real numbers x, we have [11]

Therefore,

Now on substituting (14) in (13), we find that

Differentiating with respect to t and evaluating at, we get the mean of the r-th when

and when that

as obtained by [12] for exponential distribution at.

1) Putting, in (11) and (17), the explicit formula for marginal moment generating function and mean of order statistics from Erlang-truncated exponential distribution can be obtained as

and

where

.

2) Setting in (16) and (18), the results for upper records from Erlang-truncated exponential distribution may be obtained in the form

and

as obtained by [13] for exponential distribution at .

A recurrence relation for marginal moment generating function for from (1) can be obtained in the following theorem.

Theorem 2.1 For the distribution given in (1) and for

Proof [10] has shown that for a positive integer,

On substituting for from (6) in (20) and simplifying the resulting expression, we find that

Differentiating both the sides of (21) j times with respect to t, we get

The recurrence relation in (19) is derived simply by rewriting the above equation.

At in (19), we obtain the recurrence relations for moments of from Erlang-truncated exponential distribution in the form

Remark 2.1 Putting, in (19) and (22), we can get the relations for marginal moment generating function and moments of order statistics for Erlang-truncated exponential distribution as

and

Remark 2.2 Setting and in (19) and (22), relations for record values can be obtained as

and

for

Remark 2.3 At, in (22), the result for single moments of gos obtained by [2] for exponential distribution is deduced.

Before coming to the main results we shall prove the following Lemmas.

Lemma 3.1 For the Erlang-truncated exponential distribution as given in (1) and non-negative integers a, b and c with,

where

Proof From (24), we have

where

On substituting for from (2) in (26), we get

.

Upon substituting this expression for in (25) and then integrating the resulting expression, we establish the result given in (23).

Lemma 3.2 For the distribution as given in (1) and any non-negative integers a, b and c,

where is as given in (24).

Proof Expanding binomially in (24) after noting that

, we get

Making use of Lemma 3.1, we establish the result given in (27).

When, asso after applying L’Hospital rule and (15), (28) can be proved on the lines of (16).

Theorem 3.1 For Erlang-truncated exponential distribution as given in (1) and for

Proof From (5), we have

upon using the relation (6). Now expanding binomially in (31), we get

Making use of Lemma 3.2, we establish the relation given in (30).

1) Putting, in (30), the explicit formula for the joint moment generating function of order statistics of the Erlang-truncated exponential distribution can be obtained as

where

.

2) Putting in (30), we deduce the explicit expression for joint moment generating function of upper k record values for Erlang-truncated exponential distribution in view of (29) and (28) in the form

Differentiating and evaluating at, we get the product moments of when

and when that

and for

.

Making use of (6), we can derive the recurrence relations for joint moment generating function of from (5).

Theorem 3.2 For the distribution given in (1) and for

Proof [10] has shown that for and a fixed positive integer

Differentiating both the sides of (34) times with respect to and then times with respect to, we get

which, when rewritten gives the recurrence relation in (26).

At in (34), we obtain the recurrence relations for product moments of gos from Erlang-truncated exponential distribution in the form

One can also note that Theorem 2.1 can be deduced from Theorem 3.2 by letting tends to zero.

Remark 3.1 Putting, in (34) and (35), we obtain the recurrence relations for joint moment generating function and product moments of order statistics for Erlang-truncated exponential distribution in the form

and

as obtained by [14] for exponential distribution at and.

Remark 3.2 Substituting and, in (34) and (35), we get recurrence relations for joint moment generating function and product moments of upper k record values for Erlang-truncated exponential distribution.

Let be, then the conditional of given , in view of (4) and (5), is

Theorem 4.1 Suppose, for all be a distribution function of the random variable X and, , then

if and only if

.

Proof From (36), we have

By setting from (2) in (38)we obtain

where

.

Again by setting in (39), we get

and hence the relation given in (37).

To prove sufficient part, we have from (36) and (37)

where

.

Differentiating (40) both the sides with respect to, we get

or

where

and

Therefore,

which proves that