Episode #9 of the course Theory of probability by Polina Durneva
Today and tomorrow, we will talk about two continuous probability distributions: exponential and gamma distributions. It is important to note that these two distributions are very closely interrelated and can be derived from each other. In this lesson, we will mainly discuss exponential distribution, which is also associated with Poisson distribution, discussed previously.
What Is Exponential Distribution?
Exponential distribution, also called negative exponential distribution, is used when we talk about time. In Poisson distribution, we only care about events happening over a period of time. For example, knowing that a fast food restaurant has 100 customers per hour, we might want to know the probability of 120 customers coming to the restaurant during the next hour. If we model the time in between each customer coming to the restaurant, we will use exponential distribution. Having 100 customers per hour, or a new customer every 36 seconds (1 hour * 60 minutes * 60 seconds = 3,600 seconds, and then 3,600 seconds / 100 customers = 36), we might want to know the probability that a new customer will arrive in 5 seconds. In some cases, time between events can be replaced by space between objects.
The following assumptions should be held for exponential distribution (please note that the same assumptions are held for Poisson distribution):
1. The intervals over which the events occur do not overlap.
2. The events are independent.
3. The probability that more than one event happens in a very short time period is approximately 0.
Formula for Exponential Distribution
The formula for exponential distribution takes the following form: P(x) = λ * e-xλ, where λ is the rate of occurring events, e is the natural number, and x is the random variable.
Let’s use an example from the previous section to perform a calculation and better understand the concept. We have a restaurant and know that 100 customers come in every hour, meaning that a new customer arrives every 36 seconds. The rate, or λ, is 1/36, then. Let’s assume that we want to know the probability of a new customer arriving in 5 seconds, or x = 5. Then, P(x = 5) = (1/36) * e-(5 * 1/36) = 0.024 = 2.4%. As we can see, the probability that a new customer arrives in 5 seconds and not in 36 seconds is very low.
Expected Value of Exponential Distribution
The expected value, or mean, of exponential distribution can be calculated by dividing 1 by the rate λ. In our previous example, the expected value will be 1/λ = 36 seconds, which totally makes sense: Every new customer arrives every 36 seconds on average.
Variance of Exponential Distribution
Finally, the variance of exponential distribution is a bit more complicated and can be derived using the formula 1/λ2. In our example with customers arriving to a fast food restaurant, the variance would be 1/(1/36)2 = 362 = 1,296.
Well, that’s the end of our lesson for today! Tomorrow, we will finish the course by talking about Gamma distribution, which is basically derived from exponential distribution.
See you soon,
Introduction to Probability by Joseph K. Blitzstein, Jessica Hwang
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