Let p = 1% and n = 500. a. Is the Poisson distribution a...

Let p = 1% and n = 500. a. Is the Poisson distribution a reasonable approximation of the binomial distribution? \text{yes} \text{no} b. What is the mean of this binomial distribution (and the mean of the Poisson distribution used to approximate it)? mean = 5 c. What is the actual standard deviation of this binomial distribution? standard deviation (actual) = d. What is the standard deviation of the (approximate) Poisson distribution? standard deviation (approximate) = e. Are these values in reasonable agreement? \text{yes} \text{no}

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00:01 So in this question, we need to compute the following probabilities using the normal, binomial, poisson, and exponential distributions.

00:08 So for the normal poisson and exponential, we are going to assume that the mean is 17 .5.

00:14 And for the binomial distribution, we have two arguments.

00:18 We have the n and p.

00:20 But to have the same mean, we need to consider that actually n is equal to 100.

00:26 100 because when you multiply these two values, which is the average in the binomial, you're gonna have the same average as the other distributions.

00:36 So for the first one then, the x is between 10 and 20 but not included 10 or 20.

00:44 We only need to worry about including or not including a value when we are working with a discrete distribution.

00:52 So like the poisson, the binomial in this case.

00:55 So in this case we can rewrite this one as the probability of x being less or equal than 19 minus the probability of x being less or equal than 10.

01:07 And then we can find this using the cgf for each one of these distributions.

01:14 So, for example, for the normal here, using the cgf for the mean and standard deviation that we need to consider, we get that this probability difference between these two is 0 .7036.

01:32 Then for the binomial here, the difference between these two will be 0 .6809...

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