Final answer:
The cumulative distribution function (CDF) gives the probability of the random variable X being less than or equal to x. For continuous distributions, formulas can be used to find probabilities of events such as X being more than a certain number. The CDF for a discrete scenario like the number of ATMs in use would require the probabilities of each specific number of machines in use.
Step-by-step explanation:
The student is asking about the cumulative distribution function (CDF) for the number of ATMs in use at a particular branch of a bank. The CDF, denoted as P(X ≤ x), indicates the probability that the random variable X is less than or equal to a particular value x. If the random variable X represents the number of machines in use, and there are six ATMs, then the CDF would be used to find the probability of x machines being in use at any given time.
For a continuous distribution, if provided with a specific CDF such as P(X < x) = 1 -e-0.25x, we could use it to find various probabilities. For example, the probability of having more than a certain number of machines in use P(X > x) can be calculated using the formula P(X > x) = 1 − P(X < x). Thus, if we wanted to calculate the probability of more than 7 machines being in use given that there are more than 4, we would use the memoryless property of the exponential distribution, which allows us to simplify the conditional probability P(X > 7|X > 4) to P(X > 3).
When dealing with a discrete number of ATMs, we would need to know the probability of each machine being in use to construct a CDF. This distribution would be discrete as it deals with a finite number of machines (six in this case). Without further information on the individual probabilities, we cannot construct the exact CDF for the bank's ATMs scenario.