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# 数学代写|随机过程代写Stochastic Porcess代考|МАTH4740

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## 数学代写|随机过程代写Stochastic Porcess代考|Mean Residual Life (MRL)

The quantity $E{X-t \mid X>t}, t \geq 0$ is called the mean residual life (time) for the r.v. $X$.
The r.v. $X$ is said to have increasing mean residual life IMRL if $E(X)<\infty$ and $$r(t)=E{X-t \mid X>t}=\frac{\int_t^{\infty} R(x) d x}{R(t)}$$
is increasing in $t>0$.
Note that an exponential distribution with mean $\mu$ has a constant mean residual life, equal to $\mu$ and has a constant hazard rate equal to $1 / \mu$.
Analogous definitions can be given for discrete r.v.’s.
Note that $h(t)$ and $r(t)$ are connected by the relation
$$h(t)=\frac{1+r^{\prime}(t)}{r(t)} .$$

Definition: Let $X$, the life time of a unit or component, have d.f. $F(\cdot)$ and complementary d.f. $R(x)$ $=1-F(x)$. Then the component is said to be New Better (Worse) than used NBU (NWU) if
$${1-F(x+y)} \leq(\geq)[{1-F(x)}{1-F(y)}] \text {. }$$
In other words, a unit is NBU (NWU) if the survival probability of a unit aged $x$ is less (greater) than the corresponding survival probability of a new unit.

The component is said to be New Better (Worse) than Used in Expectation NBUE (NWUE) if
$$r(t)=\frac{\int_t^{\infty} R(x) d x}{R(t)} \leq(\geq) E(X)=r(0)$$
In other words, a unit is NBUE (NWUE) if the conditional expectation of the residual life of a unit at age $t$ is less (greater) than the expectation of the life time of new unit, that is, $r(t) \leq(\geq) r(0)=E(X)$.

## 数学代写|随机过程代写Stochastic Porcess代考|Further Properties

The MRL function $r(t)$ is related to the survivor function $R(t)$ by the relations
$$r(t)=\frac{1}{R(t)} \int_t^{\infty} R(x) d x$$
and
$$R(t)=\exp \left[-\int_0^t \frac{1+r^{\prime}(x)}{r(x)} d x\right]$$
A set of necessary and sufficient conditions for a function $r(t), t \geq 0$ to be an MRL function are:
(i) $r(t) \geq 0$
(ii) $r^{\prime}(t) \geq-1$
(iii) $\lim _{t \rightarrow 0}\left{\frac{r(t)}{t \ln t}\right}=0$.
[For proof refer to Swartz, G.B., The mean residual lifetime function, IEEE Trans. Rel. R-22, 108-9 (1973)]

Both the hazard rate function $h(t)$ and the mean residual life function $r(t)$ are indicators of aging; though they look similar, there are essential differences.

While the hazard rate function takes into account the immediate future (in assessing component failure or service completion), the mean residual life takes into account the complete future life. This can be seen from the following.
We have
and
\begin{aligned} h(t) R(t) & =-\frac{d}{d t} R(t) \ & =f(t), \text { when the distribution has p.d.f. } f(\cdot) \ r(t) R(t) & =\int_0^{\infty} R(x) d x . \end{aligned}

The r.h.s. of the former relation depends on the probability law at the point $t$ only, whereas the r.h.s. of the latter relation depends on the probability law at all points in the interval $(t, \infty)$. For example, consider as lifetime distribution the uniform distribution in $[a, b]$. Then
\begin{aligned} f(x) & =\frac{1}{b-a}, \quad a \leq x \leq b \ & =0, \text { otherwise. } \end{aligned}
We get
\begin{aligned} h(t) & =0, \quad 0 \leq t \leq a \ & =\frac{1}{b-t}, a<t \leq b \end{aligned}

and
\begin{aligned} r(t) & =\frac{a+b}{2}-t, \quad 0 \leq t \leq a \ & =\frac{1}{2}(b-t), a \leq t \leq b . \end{aligned}
In the interval $[0, a]$, the hazard rate function $h(t)$ does not give any indication of wearout; the actual age cannot be obtained from the hazard rate function prior to the point $a$. The MRL function $r(t)$ gives a more descriptive measure of the process of aging than the hazard rate function.

It has been shown that white IHF implies DMRL, the latter does not imply the former, so that the class of IHF distributions forms a proper subset of the class of DMRL distributions. For further results, see Muth $(1977,1980)$.

# 随机过程代写

## 数学代写|随机过程代写Stochastic Porcess代考|Mean Residual Life (MRL)

rv $X$据说有增加平均剩余寿命IMRL如果$E(X)<\infty$和$$r(t)=E{X-t \mid X>t}=\frac{\int_t^{\infty} R(x) d x}{R(t)}$$

$$h(t)=\frac{1+r^{\prime}(t)}{r(t)} .$$

## Matlab代写

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