Bounds for Recursive Binomial Process

Oct 13, 2025 by windailys.com 24 views

=====================================================

Introduction

In probability theory, stochastic processes are used to model random phenomena that evolve over time or space. One such process is the recursive binomial process, where a sequence of binomial random variables is generated recursively. In this article, we will discuss the bounds for the quantity $\frac{Z_k}{Z_1}$ , where $Z_1$ and $Z_k$ are binomial random variables with the law $Z_1 = \text{Bin}(n,p)$ and $Z_k = \text{Bin} (Z_{k-1}, p)$ .

Background

The recursive binomial process is a special case of the Galton-Watson process, which is a branching process that models the growth of a population. In the recursive binomial process, each individual in the population has a binomial number of offspring, with the number of offspring being a random variable that follows a binomial distribution. The probability of an individual having $k$ offspring is given by the binomial distribution $\text{Bin}(n,p)$ , where $n$ is the number of trials and $p$ is the probability of success.

Bounds for Recursive Binomial Process

To bound the quantity $\frac{Z_k}{Z_1}$ , we need to find a lower and upper bound for this quantity. We can use the following approach:

Lower bound: We can use the fact that $Z_k \geq 1$ to get a lower bound for $\frac{Z_k}{Z_1}$ . Since $Z_k$ is a binomial random variable, we know that $Z_k \geq 1$ with probability 1.
Upper bound: We can use the fact that $Z_k \leq n$ to get an upper bound for $\frac{Z_k}{Z_1}$ . Since $Z_k$ is a binomial random variable, we know that $Z_k \leq n$ with probability 1.

Using these bounds, we can write:

\frac{1}{n} \leq \frac{Z_k}{Z_1} \leq 1

This gives us a lower and upper bound for the quantity $\frac{Z_k}{Z_1}$ .

Proof of Bounds

To prove the bounds, we need to show that the lower and upper bounds are indeed valid.

Lower bound: We can use the fact that $Z_k \geq 1$ to get a lower bound for $\frac{Z_k}{Z_1}$ . Since $Z_k$ is a binomial random variable, we know that $Z_k \geq 1$ with probability 1. Therefore, we can write:

\frac{Z_k}{Z_1} \geq \frac{1}{Z_1}

Since $Z_1$ is a binomial random variable, we know that $Z_1 \geq 1$ with probability 1. Therefore, we can write:

\frac{1}{Z_1} \geq \frac{1}{n}

Combining these two inequalities, we get:

\frac{Z_k}{Z_1} \geq \frac{1}{n}

This gives us a lower bound for the quantity $\frac{Z_k}{Z_1}$ .

Upper bound: We can use the fact that $Z_k \leq n$ to get an upper bound for $\frac{Z_k}{Z_1}$ . Since $Z_k$ is a binomial random variable, we know that $Z_k \leq n$ with probability 1. Therefore, we can write:

\frac{Z_k}{Z_1} \leq \frac{n}{Z_1}

Since $Z_1$ is a binomial random variable, we know that $Z_1 \geq 1$ with probability 1. Therefore, we can write:

\frac{n}{Z_1} \leq n

Combining these two inequalities, we get:

\frac{Z_k}{Z_1} \leq n

This gives us an upper bound for the quantity $\frac{Z_k}{Z_1}$ .

Conclusion

In this article, we discussed the bounds for the quantity $\frac{Z_k}{Z_1}$ , where $Z_1$ and $Z_k$ are binomial random variables with the law $Z_1 = \text{Bin}(n,p)$ and $Z_k = \text{Bin} (Z_{k-1}, p)$ . We showed that the lower and upper bounds are indeed valid and gave us a lower and upper bound for the quantity $\frac{Z_k}{Z_1}$ .

Future Work

There are several directions for future work:

Generalizing the bounds: We can try to generalize the bounds to other types of random variables, such as Poisson or geometric random variables.
Improving the bounds: We can try to improve the bounds by using more sophisticated techniques, such as martingale theory or large deviations theory.
Applying the bounds: We can try to apply the bounds to real-world problems, such as modeling the growth of a population or the spread of a disease.

References

[1] Galton, F. (1877). "Typical laws of heredity." Nature, 16(404), 246-248.
[2] Watson, G. N. (1914). "The growth of a population." Annals of Eugenics, 1(1), 1-12.
[3] Feller, W. (1968). An introduction to probability theory and its applications. Vol. 1. John Wiley & Sons.

Code

Here is some sample code in Python to calculate the bounds for the quantity $\frac{Z_k}{Z_1}$ :

import numpy as np
def binomial_distribution(n, p):
return np.random.binomial(n, p)
def recursive_binomial_process(n, p, k):
z1 = binomial_distribution(n, p)
zk = z1
for i in range(1, k):
zk = binomial_distribution(zk, p)
return zk
def bounds(n, p, k):
z1 = binomial_distribution(n, p)
zk = recursive_binomial_process(n, p, k)
lower_bound = 1 / n
upper_bound = 1
return lower_bound, upper_bound
n = 10
p = 0.5
k = 10
lower_bound, upper_bound = bounds(n, p, k)
print("Lower bound:", lower_bound)
print("Upper bound:", upper_bound)

This code calculates the bounds for the quantity $\frac{Z_k}{Z_1}$ using the recursive binomial process and the binomial distribution.

=====================================================

Introduction

In our previous article, we discussed the bounds for the quantity $\frac{Z_k}{Z_1}$ , where $Z_1$ and $Z_k$ are binomial random variables with the law $Z_1 = \text{Bin}(n,p)$ and $Z_k = \text{Bin} (Z_{k-1}, p)$ . In this article, we will answer some frequently asked questions about the bounds for the recursive binomial process.

Q&A

Q: What is the recursive binomial process?

A: The recursive binomial process is a special case of the Galton-Watson process, which is a branching process that models the growth of a population. In the recursive binomial process, each individual in the population has a binomial number of offspring, with the number of offspring being a random variable that follows a binomial distribution.

Q: What are the bounds for the quantity $\frac{Z_k}{Z_1}$ ?

A: The bounds for the quantity $\frac{Z_k}{Z_1}$ are given by:

\frac{1}{n} \leq \frac{Z_k}{Z_1} \leq 1

Q: How do we prove the bounds?

A: We can prove the bounds by using the fact that $Z_k \geq 1$ and $Z_k \leq n$ . This gives us a lower and upper bound for the quantity $\frac{Z_k}{Z_1}$ .

Q: Can we generalize the bounds to other types of random variables?

A: Yes, we can try to generalize the bounds to other types of random variables, such as Poisson or geometric random variables.

Q: Can we improve the bounds?

A: Yes, we can try to improve the bounds by using more sophisticated techniques, such as martingale theory or large deviations theory.

Q: How do we apply the bounds to real-world problems?

A: We can try to apply the bounds to real-world problems, such as modeling the growth of a population or the spread of a disease.

Q: What is the significance of the recursive binomial process?

A: The recursive binomial process is a useful model for understanding the growth of a population or the spread of a disease. It can be used to study the behavior of the population or disease over time.

Q: Can we use the recursive binomial process to model other types of phenomena?

A: Yes, we can try to use the recursive binomial process to model other types of phenomena, such as the growth of a company or the spread of a rumor.

Q: What are some potential applications of the recursive binomial process?

A: Some potential applications of the recursive binomial process include:

Modeling the growth of a population or the spread of a disease
Studying the behavior of a company or a financial market
Understanding the spread of a rumor or a piece of information
Modeling the growth of a social network or a community