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 the variance of X, we need to use the properties of the Poisson random variable.
First, let's define the random variable X as the number of people who entered the bank before you. The parameter of the Poisson distribution is given by λt, where λ is the average number of people who enter the bank per minute, and t is the time interval in minutes.
Since the bank opens at 8am and you arrive at the bank at a uniformly random time between 8am and 9am, the total time interval is 60 minutes (from 8am to 9am). Therefore, the parameter of the Poisson distribution is λt = λ * 60.
Now, to find the expectation (mean) of X, denoted as E(X), we can use the formula E(X) = λt. In this case, E(X) = λ * 60.
To find the variance of X, denoted as var(X), we can use the formula var(X) = λt. In this case, var(X) = λ * 60.
However, since we do not know the value of λ, we still need to determine it.
To find λ, we can use the information that the number of people who enter the bank during a time interval of t minutes is a Poisson random variable with the parameter λt. We need to know the average number of people who enter the bank per minute.
To determine the average number of people who enter the bank per minute, we can use historical data or estimates provided. Suppose we have the average number of people who enter the bank per minute as μ. Then λ = μ / 60.
Finally, we can substitute the value of λ in the expectation and variance formulas to find E(X) and var(X).
E(X) = (μ / 60) * 60 = μ
var(X) = (μ / 60) * 60 = μ
Therefore, the expectation and variance of X are both equal to the average number of people who enter the bank per minute (μ), given the above assumptions and information.