Problem 19SE

Let X represent the number of events that are observed to occur in n units of time or space, and assume X ~ Poisson(nλ), where λ is the mean number of events that occur in one unit of time or space. Assume X is large, so that X ~ N(nλ, nλ). Follow steps (a) through (d) to derive a level 100(1 - α)% confidence interval for X. Then in part (e), you are asked to apply the result found in part (d).

a. Show that for a proportion 1 - α of all possible samples,

b. Let Show that

c. Conclude that for a proportion 1 - α of all possible samples,

d. Use the fact that to derive an expression for the level 100( 1 - α)% confidence interval for λ.

e. A 5 mL sample of a certain suspension is found to contain 300 particles. Let λ represent the mean number of particles per mL in the suspension. Find a 95% confidence interval for λ.

Solution

Step 1 of 5

Given X represents the no. of events

Xpoisson distribution with mean

And is mean of the events

Since X is large X

a) We have to show that

Here XN()

It follows that for all possible samples of proportion (1-

Multiply the inequality by -1 and add X to the inequality we will get

Hence we proved