# Log-normal distribution of an exponentially growing population

• allison_k
In summary, the conversation discusses the modeling of an exponentially growing bacterial population in a fluctuating environment using a multiplicative random walk. It explores the probability distribution P(y) for large t and calculates its mean and variance. It then considers the probability distribution of x and its mean for large t, expressing the answers in terms of the mean and variance of P(y). Finally, it discusses the growth rate of bacteria for a sloppy grad student compared to a fastidious grad student.
allison_k

## Homework Statement

Consider an exponentially growing bacterial population in a fluctuating environment.
The bacterial population can be modeled as a multiplicative random walk, starting at x = x0.
x evolves in time according to the following rules

$x(t + 1) = x(t)*(1 + ε):$ probability p;
$x(t + 1) = x(t)*(1 + δ):$ probability q;

where |ε|<< 1 and |δ|<< 1, but  they can be positive or negative.
Let y = ln x. For large t, what is the probability distribution P(y)?

What are the mean and variance of this distribution?

Now we want to know the probability distribution of x, Q(x)dx. If x is a monotonically varying function of y, then

$Q(x) dx = P(y) \left| \frac{∂y}{∂x}\right|dx$

What is the probability distribution of x for large t?
Calculate the mean of this distribution. Feel free to express your answers in terms of the mean and variance of P(y).

Finally, on average, which grows faster: the bacteria of the sloppy grad student who
sometimes forgets to feed the bacteria, and then overcompensates by giving
them extra food at later times, or the bacteria of the fastidious grad student
who maintains the same average growth rate at all times?

## The Attempt at a Solution

Really unsure how to proceed. I know I can simplify y to

y(t+1) = ln(x) + ε; probability p
y(t+1) = ln(x) + δ; probability q

But I'm no really sure how to proceed

from there. For the last part, I think it depends on the data given, but I'm not sure how to proceed.

## 1. What is a log-normal distribution?

A log-normal distribution is a probability distribution that is used to describe data that follows an exponential growth pattern. It is characterized by a skewed bell-shaped curve, with most of the data clustered towards the lower end and a long tail on the higher end.

## 2. How is a log-normal distribution related to an exponentially growing population?

A log-normal distribution is commonly used to model populations that are growing exponentially, as it allows for a wide range of values and can account for the skewed nature of population growth. It is often used in epidemiology and demography to study the spread of diseases and the growth of human populations.

## 3. What are the parameters of a log-normal distribution?

The parameters of a log-normal distribution are the mean (μ) and standard deviation (σ) of the underlying normal distribution. The mean represents the point of maximum probability, while the standard deviation determines the spread of the data.

## 4. How is a log-normal distribution different from a normal distribution?

A log-normal distribution differs from a normal distribution in that it is skewed to the right, while a normal distribution is symmetric. This means that for a log-normal distribution, the mean is not equal to the median or mode, as it would be in a normal distribution. Additionally, the range of values for a log-normal distribution is always positive, while a normal distribution can have both positive and negative values.

## 5. In what real-world scenarios is a log-normal distribution commonly used?

A log-normal distribution is frequently used to model the size of biological and environmental populations, such as the size of animal populations, the spread of diseases, and the growth of plant populations. It is also used in financial modeling and in fields such as engineering and physics to analyze data that follows an exponential growth pattern.

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