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# Exponential Distribution

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Exponential Distribution

Exponential Distribution
Recall your experience when you take an elevator. Think about usually how long it takes for the elevator to arrive. Most likely, the experience may be it frequently comes in a short while and once in a while, it may come pretty late.

Exponential Distribution
In another word, if we want to use a random variable to measure the waiting time for elevator to come, we can say that: 1. It must be continuous. 2. Smaller values have larger probability and larger values have smaller probability. Think about Geometric distribution, is there any similarities?

Exponential Distribution
Usually, exponential distribution is used to describe the time or distance until some event happens. It is in the form of: where x ≥ 0 and μ>0. μ is the mean or expected value.

Another form of exponential distribution
In this case, Then the mean or expected value is

What should exponential distribution look like

How to find probability?
We also use CDF to find probabilities under exponential distribution. Or

Example I On average, it takes about 5 minutes to get an elevator at Math building. Let X be the waiting time until the elevator arrives. (Let’s use the form with μ here) Find the pdf of X.

Example I 2. What is the probability that you will wait less than 3 minutes?

Example I 3. What is the probability that you will wait for more than 10 minutes?

Example I What is the probability that you will wait for more than 7 minutes?

Example I Given that you already wait for more than 3 minutes, what is the probability that you will wait for more than 10 minutes?

Lack of memory That is a very interesting and useful property for exponential distribution. It is called “Memorylessness” or simply “Lack of memory”. In mathematical form: Therefore, P(wait more than 10 minutes| wait more than 3 minutes)=P(wait more than 7+3 minutes| wait more than 3 minutes)=P(wait more than 7 minutes)

Mean and variance of exponential distribution
E(X)= μ or Var(X)= μ2 or

Relationship between Exponential and Poisson
Poisson is a discrete random variable that measures the number of occurrence of some given event over a specific interval (time, distance) Exponential describes the length of the interval between occurrence.

Relationship between Exponential and Poisson
Example II: A storekeeper estimated that on average, there are 10 customers visiting his store between 10am and 12pm everyday. However, it has been more than 30 minutes since the last customer visited. What is the probability for that?

Relationship between Exponential and Poisson
If we know that there are on average 10 customers visiting a store within 2-hour interval, then the average time between customers’ arrival is: 120/10=12 minutes. Therefore, the time interval between customer visits follows an exponential distribution with mean=12 minutes.

Relationship between Exponential and Poisson
Given that the storekeeper has not got any customers for more than 30 minutes, what is the probability that the storekeeper will still have no customer for another 15 minutes or more?

Finding Percentiles Given a random variable X, a percentile means a specific value of X, say x0. Usually, when we say p-th percentile, we mean there is a value x0 such that p% of the values of X fall below x0. A special case is the median, which is 50-th percentile. That means 50% of the values of X fall below it.

Finding Percentiles We have learned that given some data points, how to find the percentiles. In those cases, the number of data points is finite or limited. Now we turn to a different question, that is, to look for a percentile for a continuous random variable X, with NO data points given but there are infinitely many possible values. For example, if X~Uniform(0, 20), what is the median, 25%, 75% percentile?

Finding Percentile X~Uniform(0, 20), what is the median, 25%, 75% percentile? We can work out this kind of problem in the form of solving an equation. Let x0 be the median, then P(X<x0)=0.5. (hint: 1. think about the shape of the pdf of a uniform random variable; 2. how do we find probability for a continuous random variable?)

Finding Percentiles Also, how do we find the 25th and 75th percentile of X?

Finding Percentiles What is the mean of X?
Compare the mean and median of X, what can we find? Can we tell that from the shape of the pdf of X?

Finding percentiles Another example:
Y~Exp(5), find the median, 25th and 75th percentile.

Finding percentiles Compare the mean and median of Y, are they the same? Why?

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