Data 605 Discussion 7

Page 200, Question 20

An advertiser drops 10,000 leaflets on a city which has 2000 blocks. Assume that each leaflet has an equal chance of landing on each block. What is the probability that a particular block will receive no leaflets?

We use the Poisson distribution because:

• number of times an event happens occurs in fixed interval of time
• event happens with a known constant rate
• independent since the last event
• randomly distributed

Poisson Distribution: \(P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}\)

Expected Value; \(\mu = \lambda\)

Standard Deviation: \(\sigma = \sqrt{\lambda}\)

Poisson Distribution: \(P(X = k) = \frac{5^k e^{-5}}{0!}\)

Expected Value: \(\mu = 5\)

Standard Deviation: \(\sigma = \sqrt{5}\)

# Rate
lambda <- 5

# Probability of a block receiving exactly k leaflets
k <- 0
probability_k_leaflets <- round(dpois(k, lambda), 4)
expected_value <- lambda
standard_deviation <- round(sqrt(lambda), 4)

# Results
print(paste("Probability of a block receiving", k, "leaflets:", probability_k_leaflets))

## [1] "Probability of a block receiving 0 leaflets: 0.0067"

print(paste("Expected value:", expected_value))

## [1] "Expected value: 5"

print(paste("Standard deviation:", standard_deviation))

## [1] "Standard deviation: 2.2361"