# Poisson variable w/ uni. dist. parameter

• shaggymoods
In summary, we have a problem where X follows a Poisson distribution with parameter lambda, which in turn follows a uniform distribution between 0 and 5. We are asked to find the probability that X is greater than or equal to 3. To solve this, we can use the Poisson probability formula and the given information about lambda to find the probability that X is less than or equal to 2, and then subtract this from 1 to get the desired probability.
shaggymoods

## Homework Statement

Let
$X\sim Poi(\lambda)$
and assume
$\lambda\sim Uni(0,5)$

Q: Find
$\mathbb{P}\{X \geq 3\}$

## Homework Equations

For a Poisson r.v. with parameter lambda,
$\mathbb{P}\{X = k\}=\frac{\lambda^{k}e^{-\lambda}}{k!}$

and the probability that lambda is in the interval (0,5) is 1/5 and 0 otherwise.

## The Attempt at a Solution

I know that I first need to write out P(X<=2) and use the fact that P(X>=3)=1 - P(X<=2). However, how do I use the fact that lambda is uniform over (0,5)? I can't seem to think about it correctly.

Last edited:
Condition on $$\lambda$$.

## What is a Poisson variable?

A Poisson variable is a type of discrete random variable that represents the number of occurrences of a specific event within a given time or space. It is often used to model rare events or events that occur randomly and independently of each other.

## What is a uniform distribution parameter?

A uniform distribution parameter is a value that determines the shape and characteristics of a uniform distribution. In the context of a Poisson variable, it represents the average number of occurrences of the event per unit of time or space.

## How is a Poisson variable with a uniform distribution parameter calculated?

The probability mass function for a Poisson variable with a uniform distribution parameter is calculated using the formula P(X=k) = (e^-λ * λ^k) / k!, where λ is the uniform distribution parameter and k is the number of occurrences of the event.

## What are some real-life examples of Poisson variables with a uniform distribution parameter?

Some examples include the number of car accidents in a day, the number of customers at a store within a certain time period, and the number of hurricanes in a given year. These events occur randomly and independently of each other, making them suitable to be modeled with a Poisson variable.

## What are the limitations of using a Poisson variable with a uniform distribution parameter?

One limitation is that it assumes a constant rate of occurrence for the event being modeled. In real-life scenarios, this may not always be the case. Additionally, it may not accurately model events that are not rare or events that are dependent on each other.

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