# Questions tagged [renewal-processes]

Suppose $\{X_i\}_{i = 1}^\infty$ are i.i.d random variables, such that $P(X_1 > 0) = 1$. Then the corresponding renewal process is $\nu(t) = \max\{n \in \mathbb{N} | \Sigma_{i = 1}^n X_i \leq t\}$. Here $t \in \mathbb{R}_+$.

40 questions
Filter by
Sorted by
Tagged with
1k views

19 views

### Is the renewal function really a distribution function?

I often read that the renewal function $R(t)=\mathbb{E}(N_t)$ is a distribution function. But shouldn't a distribution function (CDF) be smaller than $1$?
38 views

27 views

### Conditional Expectation in a Renewal Process

My professor wrote the following statement down as a starting point for a problem that I'm working on, and I don't quite see where it's coming from. We are told that $\{N(t); t \ge 0\}$ is a simple ...
169 views

### renewal process question

The question and answer below is related to the renewal process. I'm curious how the "I" changed to "H(T)" as indicated by the yellow boxes. Thanks for spending the time to look over my newbie ...
70 views

### renewal equation for $\sim U(0,1)$ interarrivals

renewal equation for $\sim U(0,1)$ interarrivals should be $m(t)=t+\int_0^t{m(t-s)f(s)ds}$ how can this be solved? can I make substitution $y=t-s$ to get $m(t)=t+\int_0^t{m(y)f(y)dy}$ if all I ...
99 views

### Waiting time of two independent processes

Suppose that we have two independent alternating renewal processes such that both alternate between states "0" and "1" independently. The amount of time each of them is in state "1" and state "0" ...
122 views

### Joint distribution of consecutive renewal times

Consider a discrete analog to the Poisson process. Let the sequence $X_i$ be independent geometrically (with parameter $p$) distributed random variables that signify the inter arrival times of events. ...
9 views

### How does this problem relate to renewal processes?

You plan an car trip in which half the trip (measured in distance) will be driving on highways and the other half will be driving in the city. If your car’s estimated fuel economy is 40 m.p.g on the ...
80 views

76 views

### renewal processes

Let $X_1, X_2,...$ be a discrete renewal process, in which $X_i$ denotes the time in between renewals with distribution: $Pr(X_i=1)=p$ and $Pr(X_i=2)=q=1-p.$ I want to show that the renewal ...
53 views

### Renewal Processes (Rolling a Fair Die)

What is the expected number of rolls until you see a 6 immediately followed by a 2? Let $D_i$ = number observed on the $i^{th}$ roll Assume $T_0=0$ Now let $T_1$ = min{$i\ge2$:$D_{i-1}=6$, $D_i=2$}...
### Calculate Expectation of points in a homogenous poission process with parameter $\alpha$ as a renewal process?
If a poisson process $N$ on $[0, \infty )$ has rate $\alpha$ (ie $E N(A)=\alpha m(A)$, $m$ lebesgue measure ) can its points be represented as occurences in a renewal process with interarrival ...