Which of the following statements describes the relationship...

Name: which of the following statements describes the relationship between the poisson distribution and the binomial distribution multiple choice question it is a form of the binomial distribution 03433
Uploaded: 2023-11-27T09:25:52-08:00
Duration: 4 min 14 s
Channel: Kirsty Gledhill
Description: which of the following statements describes the relationship between the poisson distribution and the binomial distribution multiple choice question it is a form of the binomial distribution 03433

Which of the following statements describes the relationship between the Poisson distribution and the binomial distribution? Multiple choice question. It is a form of the binomial distribution that occurs when the sample size (n) is very large, and the probability of success (π) is close to 0.5. It is a limiting form of the binomial distribution when the probability of success (π) is very small and the sample (n) is very large. It is a limiting form of the binomial distribution when the sample size (n) is a large part of the population (N).

Transcript

00:01 Okay, so the relationship between the poisson and binomial distributions is that the poisson distribution is a limiting case of the binomial distribution when n is large and p is small.

00:26 So a good rule of thumb is if n is bigger than or equal to 100 and np is less than or equal to 10, part c asks us to derive this relationship and so that's fine.

00:41 So if we call x a binomial variable whose parameters are n and p and we say that the expectation of x we're going to call what we expect the expectation of the poisson variable to be, so we'll call that lambda.

00:57 We know the expectation of x for a binomial variable is just np, so we can see that our binomial parameter p is just given by lambda over n.

01:06 The probability then that x takes some value little x using our binomial formula is n choose little x times p which is lambda over n to the x times 1 minus p to the n minus x.

01:28 And we can see that this is just equal to n times n minus 1 times n minus x plus 1 over x factorial times lambda over n to the x 1 minus lambda over n to the n minus x.

01:49 And then we can see that here lambda over n to the x we can write as this and n to the x is just x products of n and up here we have x different products.

02:02 So what we're going to do is we're going to rewrite this we're going to bring the x factorial over this way and the n to the x over this way to write n over n times n minus 1 over n times dot dot dot times n minus x plus 1 over n times lambda to the x over x factorial times 1 minus lambda over n and then we'll write this bit a little bit separately.

02:31 So we'll write out to the n and then 1 minus lambda over n to the minus x...

Question

Please give Ace some feedback