The time required to pass through security screening at the...

The time required to pass through security screening at the airport can be annoying to travelers. The mean wait time during peak periods at Cincinnati/Northern Kentucky International Airport is 12.1 minutes (The Cincinnati Enquirer, February 2,2006 ). Assume the time to pass through security screening follows an exponential distribution. a. What is the probability it will take less than 10 minutes to pass through security screening during a peak period? b. What is the probability it will take more than 20 minutes to pass through security screening during a peak period? c. What is the probability it will take between 10 and 20 minutes to pass through security screening during a peak period? d. It is 8: 00 a.n. (a peak period) and you just entered the security line. To catch your plane you must be at the gate within 30 minutes. If it takes 12 minutes from the time you clear security until you reach your gate, what is the probability you will miss your flight?

The time required to pass through security screening at the airport can be annoying to travelers. The mean wait time during peak periods at Cincinnati/Northern Kentucky International Airport is 12.1 minutes (The Cincinnati Enquirer, February 2,2006 ). Assume the time to pass through security screening follows an exponential distribution.
a. What is the probability it will take less than 10 minutes to pass through security screening during a peak period?
b. What is the probability it will take more than 20 minutes to pass through security screening during a peak period?
c. What is the probability it will take between 10 and 20 minutes to pass through security screening during a peak period?
d. It is 8: 00 a.n. (a peak period) and you just entered the security line. To catch your plane you must be at the gate within 30 minutes. If it takes 12 minutes from the time you clear security until you reach your gate, what is the probability you will miss your flight?

Key Concepts

Cumulative Distribution Function (CDF) and Tail Probabilities

The CDF of the exponential distribution, given by F(x) = 1 ? e^(??x), is used to determine the probability that the random variable is less than or equal to a particular value. Conversely, the tail probability, or the probability that the variable exceeds a certain value, is found using P(X > x) = e^(??x).

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 memoryless property, meaning the probability of an event occurring in the future is independent of the past, and by a constant hazard rate.

Rate Parameter and Mean Relationship

In the exponential distribution, the rate parameter (?) is the reciprocal of the mean. Given a known average time, this parameter defines the distribution and is essential for calculating probabilities via the probability density function or cumulative distribution function.

Interval Probabilities

Probabilities involving a range of values are calculated by taking the difference between the CDF values at the endpoints of the interval. This provides the probability that the random variable falls within a specified range, a key concept when analyzing waiting times or time durations.

Transcript

00:01 From the problem, we know that the average waiting time at the airport is 12 .1 minute, and we are to assume that the wait time follows an exponential distribution.

00:20 For an exponential distribution, the probability density function is giving us 1 over the mean times e to the power negative 1 over the mean x.

00:34 So for a mean of 12 .1, you have the probability density function as 1 over the mean.

00:42 12 .1 times e with the power negative 1 over 12 .1 x.

00:50 Now for our first question we'll find the probability that it takes less than 10 minutes to pass through the security checkpoint.

01:00 So the probability that x is less than or equal to 10 and this is equal to 1 minus e to the power negative x not divided by the mean.

01:14 Therefore we have 1 minus e to the power negative x not divided by the mean therefore we have 1 minus e to the power negative 10 divided by 12 .1 minus and this gives us 0 .4 there is 76 and simply finding this gives us 0 .5624 for the next question we have to find the probability that it takes more than 10 minutes to pass to the security checkpoint you can represent this as the probability that x is greater than or equal to 20 for this, we are going to find the integral of the expression or the function 1 over 12 .1 times e raised the power negative 1 over 12 .1 d x as we integrate from 20 to infinity.

02:22 Now when we integrate the function we have negative e to the power negative 1 over 12 .1 x as we integrate from 20 to infinity.

02:35 We're going to substitute infinity and 20 in place of x.

02:41 Therefore we have negative e to the power negative 1 over 12 .1 times infinity minus negative e therefore we have plus e based the power negative 1 over 12 .1 times 20.

03:02 So this gives us negative e based to the power this whole thing becomes negative infinity plus e.

03:11 With the power negative 20 over 12 .1 let's rewrite this as e with the power negative 20 over 12 .1 minus e with the power negative infinity so this expression is equal to 0 .1915 minus e to the power negative infinity equals 0 so our answer here is 0 .1915 for the next problem, i'm going to find the probability that it takes between 10 and 20 minutes to go through the security checkpoint.

04:04 So we can represent this as the probability that 10 is less than or equal to x, which is less than or equal to 20...

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