# Questions tagged [birth-death-process]

This tag is for questions about birth and death processes. These processes are a special case of the continuous-time Markov process where the state transitions are of only two types: "births", which increase the state variable by one and "deaths", which decrease the state by one and they are used to model the size of a population, queuing systems, the evolution of bacteria, the number of people with a disease within a population etc.

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### Probability of Machine Failing with 5 Engines

The other day I posted this math question Coin Toss Games with Changing Coin Probabilities but I think it was too long. I am now trying to make a shortened and simpler version. Suppose there is a ...
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### Likelihood for Births and Deaths?

I am trying to understand the general logic on how to write the Likelihood Function of a Stochastic Process. In some situations, I think I can understand how this is done. For example, suppose $X_t$ ...
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1 vote
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### Relationship Between Poisson Process and Birth and Death Process

I am learning about Birth and Death (Stochastic) Processes (https://en.wikipedia.org/wiki/Birth%E2%80%93death_process): When a birth occurs, the process goes from state $n$ to $n + 1$. When a death ...
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### Formula for $\pi_n$ for a Birth-Death Process Stationary Distribution

Consider a birth-death process with parameters $\lambda_n = 2 \cdot (6-n)$ and $\mu_n = 3.5 \cdot n$ where $n = 0,1,2,\dots,6$. Find the stationary distribution (as a function of n). My attempt is ...
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1 vote
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### Stationary distribution of a birth-death process

Consider a birth-death process with constant parameters $\lambda_n = 4$ and $\mu_n = 5$ with $n = 0,1,2,\dots$. Find the stationary probability (as a function of n) My attempt so far is using the ...
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### Finding the Variance of Waiting Time of Pure Birth Process

$\{X(t), t ≥ 0 \}$ is a pure birth process with $X(0) = 0$. Its birth rates $λk$ are such that $λk=8−k$ for $0≤k<8$ and $λk=1$ for $k≥8$. Let $Wk =min\{t:X(t)=k\}$ be the waiting times. Find the ...
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### Birth rate of a dependent birth process?

Any two individuals in a population can mate with each other producing one offspring in an exponential time with parameter $\lambda$. At time $t=0$ there are just 2 individuals ($X_0=2$). Find the ...
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### Pure birth process with $\lambda_n =n\lambda$

In pure birth process we assume that $\lambda_n=n\lambda$ Write down system of equations that describe that process if $P(X(0)=1)=1$ Solve equations that describe $P_1, P_2, P_3$ and guess formula ...
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### Birth and death process - example transition probabilities

Suppose we have a birth-death process with state space S = $\{1,2\}$, $\lambda_1 = 2$ and $\mu_2 = 1$. Find transition probabilities. I have drawn how my process looks like: ...
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