Exponential Probability Between 12: 00 PM and 1: 00 PM, cars...

Exponential Probability Between 12: 00 PM and 1: 00 PM, cars arrive at Citibank's drive-thru at the rate of 6 cars per hour $(0.1$ car per minute). The following formula from probability can be used to determine the probability that a car will arrive within $t$ minutes of 12: 00 PM: $$F(t)=1-e^{-0.1 t}$$ (a) Determine the probability that a car will arrive within 10 minutes of 12: 00 PM (that is, before 12: 10 PM). (b) Determine the probability that a car will arrive within 40 minutes of 12: 00 PM (before 12: 40 PM ). (c) What value does $F$ approach as $t$ becomes unbounded in the positive direction? (d) Graph $F$ using a graphing utility. (e) Using INTERSECT, determine how many minutes are needed for the probability to reach $50 \%$.

Exponential Probability Between 12: 00 PM and 1: 00 PM, cars arrive at Citibank's drive-thru at the rate of 6 cars per hour $(0.1$ car per minute). The following formula from probability can be used to determine the probability that a car will arrive within $t$ minutes of 12: 00 PM: $$F(t)=1-e^{-0.1 t}$$
(a) Determine the probability that a car will arrive within 10 minutes of 12: 00 PM (that is, before 12: 10 PM).
(b) Determine the probability that a car will arrive within 40 minutes of 12: 00 PM (before 12: 40 PM ).
(c) What value does $F$ approach as $t$ becomes unbounded in the positive direction?
(d) Graph $F$ using a graphing utility.
(e) Using INTERSECT, determine how many minutes are needed for the probability to reach $50 \%$.

Key Concepts

Inverting the Exponential CDF for Quantiles

Calculating quantiles or percentiles involves inverting the CDF, a process often used to determine the time by which a certain percentage of events are expected to have occurred. This is achieved by setting the CDF equal to the desired probability and solving for t. In the context of an exponential distribution, this inverse operation is crucial for answering questions about how long one must wait for an event to occur with a specified probability.

Exponential Distribution

The exponential distribution is a continuous probability distribution used to model the time between events in a Poisson process. It is characterized by its constant hazard rate, meaning that the probability of an event occurring in a small time interval is constant regardless of how much time has passed, which makes it suitable for modeling events such as customer arrivals or machine failures.

Rate Parameter (?)

The rate parameter, often denoted by ? (lambda), represents the average number of events occurring per unit time in an exponential distribution. It is crucial because it determines the behavior of the distribution; a higher rate parameter leads to more frequent events and a steeper cumulative distribution function. In the exponential model, ? is used directly in both the probability density function and the cumulative distribution function to scale the time variable.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a probability distribution gives the probability that a random variable will take a value less than or equal to a specific value. For the exponential distribution, the CDF is expressed as F(t) = 1 - e^(-?t). This function is useful for determining the probability that an event occurs within a given time period and is fundamental to many applications, including reliability analysis and waiting time problems.

Limit Behavior of the CDF

In probability theory, the limit behavior of the CDF as the variable approaches infinity indicates the total probability accumulated from all possible outcomes. For an exponential distribution, as time t approaches infinity, the CDF approaches 1. This reflects the certainty that an event will eventually occur if you wait long enough, which is a key property in describing the distribution's long-term behavior.

Graphical Analysis of Distribution Functions

Graphical representations, such as plotting the CDF, are important for visualizing how probabilities accumulate over time. A graph of the exponential CDF typically shows a rapid increase at first, followed by a gradual leveling off as the probability approaches 1. The shape of this graph helps in understanding the likelihood of events occurring within different time intervals.

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