Show that the Poisson Mass Function has the same value for λ and λ-1

In summary: The rest is just algebra.In summary, the Poisson Mass Function has the same value at λ (i.e. the average) and at (λ-1). To prove this, one can let p(x-1)=p(x) and solve for x, as x-1 and x will have the same value for p. This can be seen in the given graph, where the parameter λ is fixed and p(x) peaks at two consecutive integers with the same value before dropping down. Algebraically, this can be shown by setting p(x-1)=p(x) and solving for x, which results in x!/(x-1)!=x. Therefore, when x=λ, p(x|λ) = p(x-
  • #1
Dakarai
10
0

Homework Statement


Show that, quite generally, the Poisson Mass Function has the same value at λ (i.e. the average) and at (λ-1).

Homework Equations


The Poisson Mass Function is
p(x) = [e^(-λ) * λ^(x)]/(x!)

The Attempt at a Solution


I started out by plugging in λ-1 into the equation to get p(x)= [e^(-λ-1) * (λ-1)^(x)]/(x!) but now I've gotten stuck. I'm not sure how to progress and prove this other than plug in different x and λ values. I've seen the graphs and such that indicate that λ and λ-1 share the same value, but I just can't figure out how to mathematically prove it.
 
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  • #2
If I'm not mistaken, ## \lambda ## does not change in what you are trying to show. One thing you can do is let ## p(x-1)=p(x) ## and solve for ## x ##.
 
  • #3
Thank you for the advice. I'm very confused as to how this would actually show that p(x|λ) = p(x|λ-1).
 
  • #4
This one is quite simple I think. ## e ^ {-\lambda} * \lambda^ {(x-1)}/(x-1)!=e ^ {-\lambda} * \lambda^x/x! ## . A little algebra and you have the answer. (Note x is an integer here).
 
  • #5
Oh, I see where I'm confused. I'm not looking to solve for x-1, but rather λ-1. x remains x for this equation.
 
  • #6
Charles Link said:
If I'm not mistaken, ## \lambda ## does not change in what you are trying to show. One thing you can do is let ## p(x-1)=p(x) ## and solve for ## x ##.

I reread this, and so I've attached the homework question (and the mentioned graph) to better help describe what I'm getting at. I'm very lost how to prove this.http://imgur.com/NyuuHmT
 
  • #7
Capture.PNG
 
  • #8
## \lambda ## is a parameter that is pre-selected that gives the shape of the curve. For a given ## \lambda ## there will be two consecutive integers, x-1 and x that have the same value for ## p ## i.e. ## p(x-1)=p(x) ##. That equation, in post #4 is easy to solve. ## x!/(x-1)!=x ##. I could show you the rest, but the Forum rules want the OP to do some of the work. When you get this answer for ## x ## it tells you where this "peak" occurs, i.e. for a given pre-selected ## \lambda ##, where the ## p ## function levels off with two equal values before it drops back down. If ## p(x) ## were a continuous function, you would set ## dp(x)/dx=0 ## to find this peak. Instead, ## x ## has only integer values, so you set ## p ## equal for two consecutive integers.
 
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  • #9
Dakarai said:
Thank you for the advice. I'm very confused as to how this would actually show that p(x|λ) = p(x|λ-1).

No: look at the diagram you supplied. The parameter ##\lambda## is fixed, and you are being asked to show that ##p(x|\lambda) = p(x-1|\lambda)## when ##x = \lambda##. That's all.
 
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What is the Poisson Mass Function?

The Poisson Mass Function is a mathematical function that represents the probability of a certain number of events occurring within a specific interval of time or space, given the average rate of occurrence of those events.

What is the significance of λ in the Poisson Mass Function?

λ, also known as the rate parameter, represents the average rate of occurrence of the events being studied.

How is the Poisson Mass Function calculated?

The Poisson Mass Function is calculated using the formula P(x;λ) = (e^-λ)(λ^x)/x!, where x is the number of events and λ is the rate parameter.

Why does the Poisson Mass Function have the same value for λ and λ-1?

This is because when λ is decreased by 1, the value of P(x;λ) decreases by a factor of λ/(λ-1). However, at the same time, the value of (λ^x)/x! increases by a factor of (λ-1)/λ. These two factors cancel each other out, resulting in the same value for P(x;λ) as for P(x;λ-1).

What is the practical application of understanding the Poisson Mass Function?

The Poisson Mass Function is commonly used in various fields such as statistics, physics, and biology to model the probability of rare events occurring. It can help in predicting the likelihood of future events and making informed decisions based on that information.

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