# What is the name and application of this probability distribution

• Lurco
In summary, the conversation is discussing a probability distribution and trying to determine its name and applications. It is determined to be the Poisson distribution, which is defined by a formula involving a rate parameter and the number of events observed. The conversation also mentions a normalization factor and confirms that the formula does not vary with k.
Lurco
Hi.

In my homework I've encountered a discrete probability distribution of this form:

$$f(k,\lambda)=N \frac{\lambda^k}{k!}$$

$$k$$ is the variable, and $$\lambda$$ is a parameter. I'm curious what is this distribution - what's its name and where can it be applied. I will be grateful for, for example, redirecting me to the proper wikipedia article. Thanks!

Lurco said:
Hi.

In my homework I've encountered a discrete probability distribution of this form:

$f(k;\lambda) = \frac{\lambda ^{k} e^{\lambda}}{k!}$

$$k$$ is the variable, and $$\lambda$$ is a parameter. I'm curious what is this distribution - what's its name and where can it be applied. I will be grateful for, for example, redirecting me to the proper wikipedia article. Thanks!

Are you sure you copied the formula correctly? The Poisson distribution is defined by:

$f(k;\lambda) = \frac{\lambda ^{k} e^{-\lambda}}{k!}$

where $\lambda$ is the rate parameter (expected number of events per unit time), and k is the number of events observed.

In evaluating Poisson noise the question becomes $P(k=N_t)$ but your formula still doesn't look right since it lacks the exponential term.

Last edited by a moderator:
I think the number N here is used as a normlization factor.

SW VandeCarr said:
Are you sure you copied the formula correctly? The Poisson distribution is defined by:

$f(k;\lambda) = \frac{\lambda ^{k} e^{-k}}{k!}$

where $\lambda$ is the rate parameter (expected number of events per unit time), and k is the number of events observed.

In evaluating Poisson noise the question becomes $P(k=N_t)$ but your formula still doesn't look right since it lacks the exponential term.

shuxue1985 said:
I think the number N here is used as a normlization factor.

[EDIT]: After reading the wiki page, yes the value depends on lambda not k.

Can't believe I've used this pdf so many times and forgotten it!

Last edited:
Thank you all for responding. Yes, the number N stand for the normalization constant, and in the wikipedia article posted by micromass the exponent is exactly the normalization:
$$e^{-\lambda},$$

so i does not vary with k.

## What is the name of this probability distribution?

The name of this probability distribution is the normal distribution.

## What is the application of this probability distribution?

The normal distribution is commonly used in statistics to analyze and model continuous data. It is also used in many scientific fields, such as physics, engineering, and finance, to describe real-world phenomena.

## How is this probability distribution defined?

The normal distribution is defined by two parameters: the mean, which represents the center of the distribution, and the standard deviation, which measures the spread or variability of the data.

## What are the characteristics of this probability distribution?

The normal distribution is symmetric, bell-shaped, and continuous. It is also known as a Gaussian distribution and follows the 68-95-99.7 rule, where approximately 68% of the data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

## What are some real-world examples of this probability distribution?

The normal distribution can be found in many real-world scenarios, such as heights and weights of individuals, IQ scores, measurement errors, and stock prices. It is also used in quality control to assess the variability of a product's characteristics.

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