PG 201 Question 26

Feller discusses the statistics of flying bomb hits in an area in the south of London during the Second World War. The area in question was divided into 24 × 24 = 576 small areas. The total number of hits was 537. There were 229 squares with 0 hits, 211 with 1 hit, 93 with 2 hits, 35 with 3 hits, 7 with 4 hits, and 1 with 5 or more. Assuming the hits were purely random, use the Poisson approximation to find the probability that a particular square would have exactly k hits. Compute the expected number of squares that would have 0, 1, 2, 3, 4, and 5 or more hits and compare this with the observed results.

Using a Poisson Distribution:

poisson <- function(lambda, x){
  poisson <- ((lambda^x)*(exp(-lambda)))/(factorial(x))
  return (poisson)
}

lambda <- 537/576
values <- c()
for (x in 0:5){
  temp <- poisson(lambda, x)
  value <- round(temp*576, 0)
  values <- c(values, value)
  print(value)
}
## [1] 227
## [1] 211
## [1] 99
## [1] 31
## [1] 7
## [1] 1
number <- c(0, 1, 2, 3, 4, 5)
observed <- c(229, 221, 93, 35, 7, 1)

data.frame(number, observed, values)
##   number observed values
## 1      0      229    227
## 2      1      221    211
## 3      2       93     99
## 4      3       35     31
## 5      4        7      7
## 6      5        1      1