It is known that the number of people who enter a bank during a time interval of
t minutes is a Poisson random variable with the parameter �t. The bank opens at 8am and you
arrive at the bank at uniformly random time between 8am and 9am. Let X be the number of
people who entered the bank before you. Find the expectation and the variance of X.
To find the expectation and variance of X, we need to use the properties of the Poisson distribution and the concept of conditional probability.
Let's start by finding the expectation of X.
The Poisson random variable with parameter λ follows the probability mass function:
P(X = k) = e^(-λ) * (λ^k) / k!
In this case, the parameter λ is μt, where μ represents the average number of people entering the bank per minute and t is the time interval in minutes.
Since we are interested in the number of people who entered the bank before you, X can take on values from 0 to infinity.
The expectation of a Poisson random variable is given by E(X) = λ.
In our case, λ = μt. But we need to find λ conditional on the event that you arrive at the bank between 8am and 9am.
Since you arrive at a uniformly random time between 8am and 9am, the average time between your arrival and the bank opening is 30 minutes. Therefore, t = 30.
To find the expectation of X, we can calculate E(X) = λ = μ * t = μ * 30.
However, we still need to determine μ, which represents the average number of people entering the bank per minute.
To do this, we need more information.