In analyzing hits by bombs in a past war a city was...

In analyzing hits by bombs in a past war, a city was subdivided into 539 regions, each with an area of 0.25 mi². A total of 420 bombs hit the combined area of 539 regions. The Poisson distribution applies because we are dealing with the occurrences of an event (bomb hits) over some interval (a region with an area of 0.25 mi²). Find the mean number of hits per region: mean = Find the standard deviation of hits per region: standard deviation = If a region is randomly selected, find the probability that it was hit exactly twice. (Report answer accurate to 4 decimal places.) P(X = 2) = Based on the probability found above, how many of the 539 regions are expected to be hit exactly twice? (Round answer to a whole number.) ans = If a region is randomly selected, find the probability that it was hit at most twice. (Report answer accurate to 4 decimal places.) P(X ≤ 2) =

Transcript

00:01 In this question we are given in the past wall of a city.

00:04 A total of 420 bombs hit the combined area of 539 regions, where each region's area is 0 .25 square miles.

00:15 I'm going to let a be the number of hits in the given combined area of 539 regions.

00:22 So a follows the poison distribution and the mean will be 420.

00:30 I'm going to let x be the number of hits in a given region, so one region.

00:37 So x will also follow the poison distribution since a follow the poison distribution.

00:43 The mean will be 420 divided by 5 .39.

00:50 So x follows the poison distribution with the mean of 60 over 77.

01:00 7th.

01:11 Now we want to find the mean number of heats per region that is finding the expectation of x.

01:18 Now since x follows the poison distribution, the mean will be over here which is 60 out of 77 or 0 .7792 for decimal place.

01:35 Next we want to find the standard deviation of heats per region so that is the square root of the variance of x.

01:43 Now since x follows the poison distribution, the variance will be the same as the mean.

01:52 So the standard deviation would be square root of 60 over 77 and that would be 0 .8827 4 decimal place.

02:06 Next we want to find probability of a randomly selected region was hit exactly twice.

02:12 So we're looking at probability of x equals to 2.

02:17 Now since x follows the poison distribution, probability of x equals to r where r is the number of heats in a given region will be 60 over 77 to the power of r divided by r factorial times e to power minus 60 over 77 and so for x equals to 2 just sub 2 into the r and we get this 0 .139 277 which is 0 .1393 4 decimal place.

03:32 Next, i'm going to let y be the number of regions out of 539 regions that are hit exactly twice.

03:42 So why will follow the binomial distribution where n, the number of trials will be 539 since there are 539 regions, there will be 539 trials.

03:54 In each trial, we're looking at whether a region is hit exactly twice.

03:58 And p, probability of success in a single trial, in this case probability of a region being hit exactly twice, and that will be this...

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