Suppose small aircraft arrive at a certain airport according...

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t. (Round your answers to three decimal places.) (a) What is the probability that exactly 8 small aircraft arrive during a 1-hour period? What is the probability that at least 8 small aircraft arrive during a 1-hour period? What is the probability that at least 12 small aircraft arrive during a 1-hour period? (b) What is the expected value and standard deviation of the number of small aircraft that arrive during a 105-min period? (c) What is the probability that at least 23 small aircraft arrive during a 2.5-hour period? What is the probability that at most 11 small aircraft arrive during a 2.5-hour period?

Transcript

00:01 In this question, we have been given about the number of aircrafts arriving in a certain time period and it follows the poison distribution.

00:09 So here we can say that probability of x and then here it will be mu and this is here given by e raised to the power negative ut and in the denominator it will be here x factorial and now in the numerator it will be we can say mu raised to the power x.

00:30 So with the help of this very formula and we have certain information such as the value of u is here equals to 8t and the value of alpha is here equals to 8.

00:43 So in the first part of this very question we need to find out probability of it will be here x and that is basically equals to 8.

00:53 So it is here going to come around 8 raised to the power 8 and then here it is e raised to the power 9.

01:00 Negative 8 divided with 8 factorial and it will be here equals to 0 .1395 and this will be our answer for the first part of this very question where we have found out the required value now in the first part itself we have some other quantities to find their value such as we need to calculate the probability that x where x represent the number of aircraft that arrive during a one hour period and it will be here greater than equal to 8.

01:34 So it is going to come around 1 minus probability of x less than 8 and it will be here equals to 0 .5474.

01:45 And now if we simplify it further, then let's see what value we are going to have.

01:51 Because in the first part we have something else to find out.

01:56 And that can be here written as probability of, it will will be here x greater than equals to 12 and it is here equals to 1 minus probability of it will be here x lesser than 11 and that will be equals to 0 .11192 and that's how we have find out the answer for the third thing which is asked in the first part of this very problem now let us see how we can find out the answer for the b part of this very question so in the b part of this very question so in the b we have been asked about the expected value as well as the standard deviation...

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