# Question: Suppose that f x t is the probability of getting x

Suppose that f(x, t) is the probability of getting x successes during a time interval of length t when

(i) The probability of a success during a very small time interval from t to t + △t is α ∙ △t,

(ii) The probability of more than one success during such a time interval is negligible,

(iii) The probability of a success during such a time interval does not depend on what happened prior to time t.

(a) Show that under these conditions

And hence that

(i) The probability of a success during a very small time interval from t to t + △t is α ∙ △t,

(ii) The probability of more than one success during such a time interval is negligible,

(iii) The probability of a success during such a time interval does not depend on what happened prior to time t.

(a) Show that under these conditions

And hence that

## Answer to relevant Questions

Use repeated integration by parts to show that This result is important because values of the distribution function of a Poisson random variable may thus be obtained by referring to a table of incomplete gamma functions. Use Theorem 5.9 to find the moment– generating function of Y = X – λ, where X is a random variable having the Poisson distribution with the parameter λ, and use it to verify that σ2Y = λ. Suppose that the probability is 0.63 that a car stolen in a certain Western city will be recovered. Use the computer printout of Figure 5.1 to find the probability that at least 8 of 10 cars stolen in this city will be ...(a) Use a computer program to calculate the probability that more than 12 of 80 business telephone calls last longer than five minutes if it is assumed that 10 percent of such calls last that long. (b) Can this result be ...A quality control engineer inspects a random sample of two hand– held calculators from each incoming lot of size 18 and accepts the lot if they are both in good working condition; otherwise, the entire lot is inspected ...Post your question