Question

Traffic surveys at an intersection were conducted on four different occasions, recording the number of vehicles per hour as follows: 42, 46, 60, and 38 vehicles. It is assumed that the number of vehicles follows a Poisson distribution. Historical records show that the mean rate of vehicles in the suburb is 55 vehicles per hour. Which one of the following expressions represents the posterior distribution of the mean rate of vehicles (θ) at the intersection using the historical record as an informative prior?

a) ln(p(θ∣D)) = K + 241⋅5θ + ln(θ)

b) ln(p(θ∣D)) = K + 241ln(θ) − 5θ

c) ln(p(θ∣D)) = K + 241ln(5θ) − θ

d) ln(p(θ∣D)) = K + 5θln(θ) − 241

Answer 1

The posterior distribution of the mean rate of vehicles (θ) at the intersection using the historical record as an informative prior is represented by ln(p(θ∣D)) = K + 241ln(θ) − 5θ.

Step-by-step explanation:

The posterior distribution of the mean rate of vehicles (θ) at the intersection using the historical record as an informative prior can be represented by the following expression:

ln(p(θ∣D)) = K + 241ln(θ) − 5θ

In this expression, θ represents the mean rate of vehicles at the intersection, D represents the observed data from the traffic surveys, ln() represents the natural logarithm, and K is a constant.

This expression is derived from Bayesian inference, where the prior distribution (in this case, based on historical records) is combined with the likelihood of the observed data to obtain the posterior distribution.