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Consider a renewal process with the lifetimes $X_1,X_2,\ldots$ having the probability density function $f(x)=0.125\cdot (4-x)\;$ for $\;0<x<4$

Determine the asymptotic expression for the probability distribution of excess life $\gamma_t$: $\displaystyle\lim_{t\to\infty} \mathbb{P}(\gamma_t\leq 1.45)$

Determine the limiting mean excess life: $\displaystyle\lim_{t\to\infty} \mathbb{E}[\gamma_t]$

So far I have correctly calculated that $\displaystyle\lim_{t\to\infty} \mathbb{P}(\gamma_t\leq 1.45) = \frac{3}{4}\cdot\frac{379349}{384000}$ and my attempt to calculating $\displaystyle\lim_{t\to\infty} \mathbb{E}[\gamma_t] = \frac{M(t)}{t} + \frac{1}{t}-\frac{1}{\mu} = \frac{1}{\mu} + \frac{1}{t} - \frac{1}{\mu} = \frac{1}{t} = 0$ by using the fact that $\frac{M(t)}{t} \rightarrow \frac{1}{\mu}$ as $t \rightarrow \infty$. But this answer was incorrect, I would appreciate any help in figuring this out.

Edit: $\gamma_t$ is the residual life or excess (at time t) where $\gamma_t = W_{N(t) +1} - t$ and $\gamma_t \sim Exp(\lambda)$ .$E[\gamma_t] = \frac{M(t)}{t} + \frac{1}{t} - \frac{1}{\mu}$ where $M(t)$ is the renewal function where $\frac{M(t)}{t} \rightarrow \frac{1}{\mu}$ as $t \rightarrow \infty$ and $W_{N(t)+1}$ is the first renewal time after time $t$. $\mu$ is the mean of the given distribution

waterr
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  • Please define excess life $\gamma_t$. – jdods Mar 19 '24 at 12:32
  • @jdods $\gamma_t$ is the residual life or excess (at time t) where $\gamma_t = W_{N(t)+1}-t$ and $E[\gamma_t] = \frac{M(t)}{t} + \frac{1}{t} - \frac{1}{\mu}$ where $M(t)$ is the renewal function – waterr Mar 19 '24 at 12:40
  • Please define $W_{N(t)+1}$, $M(t)$, and $\mu$. Generally it is advised to define everything or to at least say something is a standard definition. You’ll find that folks here might be able to answer your question even if they lack specific familiarity with standard terminology. So you are more likely to get an answer with everything really clearly laid out. – jdods Mar 19 '24 at 20:42
  • @jdods thank you for the tips, I have edited the question so that they are defined. Let me know if these definitions are enough and I'll add more if they are insufficient – waterr Mar 20 '24 at 12:38
  • There must be something wrong: $\gamma_t$ doesn't seem like it would just be an exponential random variable with rate $\lambda$. Please check all equations and definitions very carefully. – jdods Mar 20 '24 at 19:56

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