Final answer:
The exponential distribution is used to model the time between events in various contexts. It requires knowing the average rate of occurrence (λ) to calculate probabilities related to time intervals between these events. The rate, in turn, helps determine probabilities for events occurring within or exceeding a certain timeframe.
Step-by-step explanation:
Understanding Exponential Distribution in Various Contexts
The exponential distribution is often used to model the time between events in a Poisson process. The key parameter in an exponential distribution is the rate (λ), which is the average number of events per time unit. The probability of waiting more than a certain amount of time can be calculated using the exponential distribution formula ℓ(t) = e-λt, where ℓ(t) represents the probability of waiting longer than time 't' and 'λ' is the rate.
For example, if a website receives an average of 12 visits per hour, the rate 'λ' would be 12/60 visits per minute since there are 60 minutes in an hour. To find the probability that no visits occur in a given time frame, such as more than 10 minutes, we would substitute the given values into the formula.
Similarly, other parts of the questions involve determining probabilities and rates of events such as calls to a police station or arrivals at a store, all of which can be modeled by the exponential distribution assuming independence of events and a constant average rate.