2 The number of orders arriving at a shop during an 8-hour working day is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 25.2 )\).
- State two assumptions that are required for the Poisson model to be valid in this context.
- Find the probability that the number of orders that arrive in a randomly chosen 3-hour period is between 3 and 5 inclusive.
- Find the probability that, in two randomly chosen 1 -hour periods, exactly 1 order will arrive in one of the 1 -hour periods, and at least 2 orders will arrive in the other 1 -hour period. [4]
- The shop can only deal with a maximum of 120 orders during any 36-hour period.
Use a suitable approximating distribution to find the probability that, in a randomly chosen 36-hour period, there will be too many orders for the shop to deal with.