This tutorial covers Poisson distributions. Now a Poisson distribution gives the probability of a given number of events happening. Now it depends on having a fixed interval-- so a fixed length of time, a fixed, set amount. And then we know the average rate. So we know on average how much something fails. And then the events happen independently. Finally, it can be used when the number of trials is unknown. So even though we don't know the number of trials, because we know the average rate and have the fixed interval and the independence, then we can still use a Poisson distribution to figure out the likelihood of an event happening.
So here is the formula for a Poisson distribution. Now in this formula up here, e is a constant. It's about 2.71828. That's an approximation. The real number goes on for a lot longer. u-- or mu rather, is a mean number of successes. So on average, how often does the success occur? x is the actual number of successes. So in this particular case, what kind of success rate-- number of successes are we looking for? And then p of u colon x is the Poisson probability that exactly x successes occur when the mean number of successes is mu.
So here this is helping us to find out that probability that we were talking about before. For example, on an average day, five iPhone owners under warranty have a hardware failure. What is the likelihood that tomorrow six owners will experience failure? So we're going to use this formula to help us calculate it. Now again, e is a constant. It's 2.71828. Then the mean number of successes-- so we're going to consider the hardware failure a success. So the mean number of successes is 5. And the actual number of successes-- we want to know if tomorrow we have 6 successes, what's that going to be?
So our Poisson distribution for the actual successes of 6 when we have an average mean of 5 equals 2.71828 to the negative mean, so to the negative 5, times u-- sorry, times the mean, which is 5, to the x-- so to the 6. And then that's all divided by x. So 6 factorial.
So now we can use our calculator to do that for us. You can either enter them into the calculator, or you can find an online Poisson distribution calculator. So first, we need to have our e and raise that up to the negative mean, which in this case, was 5. And I like to use a parentheses just to ensure that the negative stays with the 5, but you don't necessarily have to.
And then we're going to be multiplying that by the mean raised to the 6, the actual x. And then we want to make sure we're dividing everything. And then again, I would use parentheses to make sure that everything is getting divided by 6 factorial.
And if you wanted to, you could write it out 6 times 5 times 4 times 3 times 2 times 1. But the web calculator will know to use the factorial. And when we do that, we get 0.146, and so forth. And we're going to round that to just be 0.146.
So this has been our calculations for the Poisson distribution. And when we do that, we get 0.146. Now that would be about 14.6%. So given that this situation occurs, we have about a 14.6% chance of having six iPhone failures in one day.
So this has been your tutorial on the Poisson distribution. Showed you how it works, what the formula is, and an example.