# Probability Question About The Poisson Probability Distribution

• Shoney45
In summary, the conversation discussed how to use the Poisson approximation to calculate the approximate standard deviation of the number of people who carry a gene that causes inherited colon cancer. The value of lambda, which represents the expected number of occurrences, was found to be equal to 5. It was also noted that the Poisson distribution only approximates the binomial distribution with a large sample size, leading to the use of the term "approximate" in the question.

#### Shoney45

Probability Question About "The Poisson Probability Distribution"

## Homework Statement

- Assume that 1 in 200 people carry the defective gene that causes inherited colon cancer. A sample of 1000 individuals is taken.

Use the Poisson approximation to calculate the appoximate standard deviation of the number of people who carry the gene.

## The Attempt at a Solution

I am honestly having a hard time even getting started with this one. I think I find Lamda by setting 1-$$e^{-\lambda1000}$$=1/2

What does it say lambda is in the definition of a Poisson distribution?

It says that Lamda is "frequently a rate per unit time or per unit area".

That's not helpful. You could describe a lot of things that way. Nothing more specific? Check wikipedia if your reference is completely lame.

It also says that lamda is equal to n*p. I think I need to use the entire population (1000) for n. But I have no idea what to use for 'p'.

Shoney45 said:
It also says that lamda is equal to n*p. I think I need to use the entire population (1000) for n. But I have no idea what to use for 'p'.

Maybe 'p' has something to do with '1 in 200'.

According to Wikipedia: "λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average 4 times per minute, and you are interested in the number of events occurring in a 10 minute interval, you would use as your model a Poisson distribution with λ = 10×4 = 40".

So it looks like lamda = 1000*1/200 = 5 (if I'm understanding this)

Shoney45 said:
According to Wikipedia: "λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average 4 times per minute, and you are interested in the number of events occurring in a 10 minute interval, you would use as your model a Poisson distribution with λ = 10×4 = 40".

So it looks like lamda = 1000*1/200 = 5 (if I'm understanding this)

You are understanding correctly.

Okay, cool. That gets me started then. I'll see what I can drum up here. Thanks for your help.

So it looks likeE(X) = V(X) = Lamda. So finding the standard deviation should be as simple as finding the square root of five right?

Shoney45 said:
So it looks likeE(X) = V(X) = Lamda. So finding the standard deviation should be as simple as finding the square root of five right?

Yes, it's that simple. Do you know why it's only the 'approximate' standard deviation for a sample of 1000?

I'm assuming it is because the square root of five is an irrational number.

Gotta get to class. Thanks for helping me. I appreciate it.

Shoney45 said:
I'm assuming it is because the square root of five is an irrational number.

No, it's because a Poisson distribution only 'approximates' a binomial distribution with a large sample size. For a binomial distribution V(X)=n*p*(1-p), but since 1-p is almost 1, you get almost the same thing. That's why they said 'approximate'. Just so you know.

Dick said:
No, it's because a Poisson distribution only 'approximates' a binomial distribution with a large sample size. For a binomial distribution V(X)=n*p*(1-p), but since 1-p is almost 1, you get almost the same thing. That's why they said 'approximate'. Just so you know.

Okay, got it. My teacher is just terrible. You have helped me more about this than pouring over this book, and every lecture that guy could ever give. Thanks again.

Sure. Nice of you to say. But I think you pore over a book, you don't pour over it. Pouring is kind of a strange image. But you are welcome!