Question

Consider a multiple birth Yule process where each member in a population has a probability \(\beta h+o(h)\) of giving birth to \(k\) new members and probability \((1-\beta h+o(h))\) of no birth in an interval of time length \(h(\beta>0, k\) positive integer). Assume that there are \(N\) members present at time \(0 .\) (a) Let \(X(t)\) be the number of splits up to time \(t\). Determine the growth behavior of \(E(X(t))\) (b) Let \(\tau_{n}\) be the time of the \(n\)th split. Find the density function of \(\tau_{n^{*}}\)

Short answer

(a) The growth behavior of E(X(t)) is linear with respect to t, given by \(E(X(t)) = N\beta t\). (b) Computing the exact expression for the density function \(f_{\tau_n}(t)\) is complicated and may not yield a simple closed-form solution in general. In practice, one can perform the computation numerically or use approximation techniques to evaluate \(f_{\tau_n}(t)\) if needed.

Step-by-step solution

Define a few notations

Here, we define the following random variables: - \(X(t)\): the number of splits up to time t - \(S_n\): the time it takes for the nth split - \(\tau_n\): the time of the nth split, given by \(\tau_n = \sum_{i=1}^{n} S_i\) Now let's find the growth behavior of \(E(X(t))\). ##Step 2: Compute the expectation of \(X(t)\)##

Compute the expectation of \(X(t)\)

We have a Yule process with k offspring. The process is essentially a continuous-time Bernoulli trial with rate β for N members. For any time interval of length h, the chance of splitting can be considered as the success probability of N independent Bernoulli trials, so the expected number of splits in a time interval of length h is given by: \(E(X(h)) = N\beta h + \mathcal{O}(h^2)\) Now, divide both sides by h and then, take the limit as h goes to zero: \(E(\frac{X(h)}{h}) = N\beta + \mathcal{O}(h)\) And finally, integrate both sides from 0 to t: \(E(X(t)) = \int_0^t N\beta ds = N\beta t\) Thus, the growth behavior of \(E(X(t))\) is linear with respect to t. Now let's find the density function of \(\tau_n\). ##Step 3: Define the random variables for birth times##

Define the random variables for birth times

We can define \(q_i\) as the time it takes for the \(i\)th member to give birth to a new member, and \(S_i = \min\{q_1, q_2, ...\, q_N\}\). So, the joint distribution function of \(S_i\)'s can be defined as: \(f_{S_1, S_2, ..., S_n}(s_1, s_2, ..., s_n) = f_{Q_1}(q_1)f_{Q_2}(q_2)...f_{Q_N}(q_N)\), where \(f_{Q_i}(q_i)\) is the probability density function of \(q_i\) and \(q_i = s_j ~(j = 1,2,\dots, i)\) for the minimum birth times. ##Step 4: Compute probability of one birth within a time interval##

Compute probability of one birth within a time interval

For any member i in the population, the probability of one birth in a time interval of length h is given by \(\beta h+o(h)\). Then, we have the probability density function for \(q_i\) as: \(f_{Q_i}(q_i) = \beta e^{-\beta q_i}\) Now, let's find the joint distribution function for \(S_1, S_2, ..., S_n\). ##Step 5: Compute joint distribution function##

Compute joint distribution function

From the definition in step 3, we have: \(f_{S_1, S_2, ..., S_n}(s_1, s_2, ..., s_n) = f_{Q_1}(q_1)f_{Q_2}(q_2)...f_{Q_N}(q_N) = \beta^n e^{-\beta(s_1 + s_2 + ... + s_n)}\) Now, let's find the probability density function of \(\tau_n\): ##Step 6: Compute probability density function of \(\tau_n\)##

Compute probability density function of \(\tau_n\)

Recall that \(\tau_n =\sum_{i=1}^n s_i\). We can now compute the probability density function of \(\tau_n\) by the formula: \(f_{\tau_n}(t) = \frac{\partial^n}{\partial t^n}F_{\tau_n}(t)\), where \(F_{\tau_n}(t)\) is the cumulative distribution function of \(\tau_n\) which can be computed through its relationship with \(F_{S_1, S_2, ..., S_n}(s_1, s_2, ..., s_n)\): \(F_{\tau_n}(t) = P(\tau_n \le t) = P(\sum_{i=1}^{n} S_i \le t)\). Integrating the joint distribution function \(f_{S_1, S_2, ..., S_n}(s_1, s_2, ..., s_n)\) for all \((s_1, s_2, ..., s_n)\) combinations such that \(\sum_{i=1}^{n} s_i \le t\), we get the cumulative distribution function \(F_{\tau_n}(t)\). Next, we differentiate \(F_{\tau_n}(t)\) n times with respect to t to obtain the required density function \(f_{\tau_n}(t)\). However, computing the exact expression for \(f_{\tau_n}(t)\) is complicated and may not yield a simple closed-form solution in general. The difficulty lies in the integration of the joint distribution function over all possible combinations of \((s_1, s_2, ..., s_n)\) such that their sum is less than or equal to t. In practice, one can perform this computation numerically or use approximation techniques to evaluate \(f_{\tau_n}(t)\) if needed.

Key concepts

Stochastic processes

In exploring mathematical phenomena, stochastic processes offer a fascinating glimpse into a world governed by random variables and their evolution over time. Imagine a sequence of random events unfolding like a story, where each chapter's outcome influences the next. This is akin to a stochastic process A classic example within this realm is the Yule process. This process unfolds over time as members within a population exhibit a certain behavior each member potentially giving birth to new members, and these births occur randomly over time In the context of our exercise, this means that each individual in the population at any given moment has a certain probability of giving birth the next event in the sequence is entirely random and contingent upon previous events. It's essential to grasp that stochastic processes allow us to model complex systems ones that are unpredictable yet adhere to certain statistical patterns. They are widely used in various fields such as finance, physics, and biology, to name a few.

Exponential distribution

Delve into the world of probability and you'll often encounter the exponential distribution, a continuous probability distribution that describes the time between events in a Poisson process. It is the continuous counterpart to the geometric distribution, which is discrete.

In simple terms, the exponential distribution helps us to determine the likelihood of a certain amount of time passing before an event occurs. For instance, if we were measuring the time until a radioactive atom decays, or, as in our exercise, the time until a new member is born in a Yule process, the exponential distribution can provide invaluable insights.

It's interesting to note that the probability density function (PDF) for the exponential distribution, often denoted as can tell us a great deal about the nature of the events we're examining. For example, it explains why we often see an initial flurry of activity followed by longer periods of waiting this is characterized by the decline in the PDF as time progresses. Such insights from the exponential distribution are instrumental in a wide array of practical applications, including risk assessment and queuing theory.

Probability density function

When we talk about the probability density function (PDF), we dive into the heart of understanding how probabilities are distributed across different outcomes in a continuous sample space.

The PDF is a function that describes the relative likelihood for a continuous random variable to occur at a given point. In layman's terms, it helps us visualize the 'landscape' of probabilities, much like a topographic map shows the terrain features of landscapes.

The key feature of a PDF is that it gives us probabilities for intervals rather than specific outcomes the area under the curve between two values on the x-axis corresponds to the probability of the variable falling between those two values.

In the Yule process we're examining, for example, the PDF of the time until the next split ( would give us a graphical representation of how likely it is for the next split to occur at any given moment. Learning to interpret and use PDFs is a foundational skill in statistics, and it provides profound insights into random phenomena. PDFs are used to identify where the values of a random variable are concentrated, where they're rare, and how they behave on average, solving many real-world problems.