May I please borrow your expertise or could anyone check if I'm on the right track please?
Consider customers arriving at a bank. The bank has $2$ types of customers – business and personal. On average, $10$ business customers arrive per hour, and $20$ personal customers. The times between arrivals of the business customers are independent of each other, and exponentially distributed, and is the same for personal customers. The two streams are independent of each other. Suppose no new customers have entered the bank in the last $5$ minutes.
1) What is the probability that no business customers arrive in the next $3$ minutes?
I'm assuming the arrival time for business customers is 60/10 = 6 customers
Thus, by using the exponential equation and also that this is a memoryless property,
$1 – e^{-3/6}$ (am I on the right track here anyone)?
2) What is the probability that no personal customers arrive in the next $3$ minutes?
Assuming the arrival time for personal customers to be 60/20 = 3 customers
Using the equation from question 1)
$1 – e^{3/3}$ (is this right)?
3) What is the probability that no customers at all arrive in the next $3$ minutes?
Assuming that this is something like $1 – ((1-e^{-3/6}) + (1-e^{-3/3}))$
by adding the no personal customer and business customer together and minus it by 1
4) What is the distribution of the time until the arrival of the next customer?
(Any hints on how I might be able to use to get the answer for this question?)
Your helps are much appreciated. Thanks,
Best Answer
We need to decide whether to measure time in hours or minutes. Say minutes. Then the mean number of business arrivals per minute is $\frac{10}{60}$. Thus we want to know the probability that an exponentially distributed random variable with parameter $\lambda=\frac{10}{60}$ is $\gt 3$. The probability that it is less than or equal to $3$ is $1-e^{-30/60}$. But the probability it is bigger than $3$ is $e^{-30/60}$.
A similar calculation for personal customers yields $e^{-60/60}$.
For no customers at all, by independence, we want the probability of no business customers times the probability of no personal customers.
As to the last question, if we want to calculate, let $W$ be the waiting time until the next customer. The probability that $W\gt w$ is the probability that $X\gt w$ and $Y\gt w$, where $X$ and $Y$ represent the waiting times for the two types of customers. Calculate this. You will recognize it as an exponential random variable. Indeed, it turns out that the distribution of $W$ can be written down immediately from the fact that the combined arrival rate is $30$ per hour.
Remark: Things will get clearer with practice, and when the relationship between the Poisson distribution and the exponential has been developed.