Exponential Probability Between 5: 00 PM and 6: 00 PM, cars...

Exponential Probability Between 5: 00 PM and 6: 00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour $(0.15$ car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within $t$ minutes of 5: 00 PM. $$ F(t)=1-e^{-0.15 t} $$ (a) Determine how many minutes are needed for the probability to reach $50 \%$. (b) Determine how many minutes are needed for the probability to reach $80 \%$.

Exponential Probability Between 5: 00 PM and 6: 00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour $(0.15$ car per minute). The following formula from statistics can be used to
determine the probability that a car will arrive within $t$ minutes of 5: 00 PM.
$$
F(t)=1-e^{-0.15 t}
$$
(a) Determine how many minutes are needed for the probability to reach $50 \%$.
(b) Determine how many minutes are needed for the probability to reach $80 \%$.

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Key Concepts

Exponential Distribution

The exponential distribution is a probability distribution that describes the time between events in a process where events occur continuously and independently at a constant average rate. It is commonly applied to model waiting times and lifespans in contexts such as reliability testing and queueing theory.

Cumulative Distribution Function (CDF)

The CDF for the exponential distribution, given by F(t) = 1 - e^(-?t), represents the probability that an event will occur within time t. This function allows users to determine the likelihood that the time between events is less than or equal to a specified value.

Poisson Process

A Poisson process is a statistical model that describes events occurring independently and at a constant average rate during a fixed period of time. The interarrival times between these independent events follow the exponential distribution, linking the two concepts.

Logarithmic Transformation

Solving for time in equations involving the exponential function often requires applying logarithms. This transformation is used to isolate the time variable, enabling the determination of the time required for a given probability threshold to be reached in an exponential setting.

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