 This is a video about using the normal distribution to approximate the Poisson distribution. Now first of all, this is extremely similar to using the normal distribution to approximate the binomial distribution. And so if you haven't done so already, I strongly recommend that you go and watch my video about the normal approximation to the binomial, which covers the theory in much more detail. But moving on, suppose that x has the Poisson distribution with parameter 12. If we draw a bar chart which shows the probability for each possible number of events, it looks something like this. And you'll notice straight away that very roughly it has the bell shape that you associate with the normal distribution. And we can be more precise about this. The mean for a Poisson distribution is the parameter lambda, the expected number of events, which in this case is 12. And the variance is also equal to lambda, so that's also 12. And if we draw the normal distribution curve with mean 12 and variance 12, you'll see that it has a very similar shape to the bar chart for the Poisson distribution. What we can say in general is that the Poisson distribution with parameter lambda, where lambda is the expected number of events, can be approximated by the normal distribution with mean lambda and variance lambda. Okay, so this is the essence of using the normal distribution to approximate the Poisson distribution. But there's something very important that you must remember whenever you do this. And that's that you need to make a continuity correction. Suppose that x has the Poisson distribution with parameter 12 and y has the normal distribution with mean 12 and variance 12. X is discrete and y is continuous. So that means that you can't say that the probability that x is 14 is about the same as the probability that y is 14. Because if you think about it, if you draw the graph of the probability density function for the random variable y, the probability that x is equal to 14 would be the area of this line. And the area of a line is zero, so the probability that y is equal to 14 would be zero. And that's why we can't say that the probability that x is 14 is about the same as the probability that y is 14. Instead, we need to make a continuity correction which involves saying that the probability that x is 14 is about the same as the probability that y is between 13.5 and 14.5. So it's very important that every time you use the normal distribution to approximate the Poisson distribution, you make a continuity correction. The other thing that you need to remember in connection with the normal approximation to the Poisson distribution is when it's valid when you're allowed to make the normal approximation. So you need to remember that you should only use the approximation when lambda is greater than 10. And let's see that working. Here's a sort of histogram for the Poisson distribution with parameter 12. We've already seen that if we superimpose the probability density function for the normal distribution with mean 12 and variance 12, they've got very similar shapes. They both have the bell shape and they're quite close together. If we look at the Poisson distribution with parameter 3 on the other hand, it's lost its bell shape. This time we've got significant positive skew and the graph looks as though it's been chopped off at the left-hand side. And this time when we try to superimpose the probability density function of the normal distribution with mean 3 and variance 3, they're less close together. Going in the other direction, if we increase the parameter lambda so that we look at the Poisson distribution with parameter 100, things look much better. This time we've got a really good looking bell shaped curve. And sure enough, if we superimpose the probability density function for the normal distribution with mean 100 and variance 100, the match is now really very close indeed. So what you need to remember is that the approximation works better, the larger the value of lambda, and you should only make it at all if lambda is greater than 10. Okay, so let's look at an example to see how we can use the normal approximation to the Poisson distribution. Suppose that x is a random variable with the Poisson distribution where the expected number of events is 85, and we want to know the approximate value of the probability that x is greater than 80. So we use the approximating random variable y, which has the normal distribution with parameters 85 and 85, where the mean is 85 and where the variance is 85. Now you have to think very carefully about what probability we need to calculate. We're asked for the probability that x is greater than 80, and that means that x will be 81, 82, and so on. So the cutoff here is somewhere between 80 and 81, and that means that to find the probability that x is greater than 80, we have to say that that's about the same as the probability that y is greater than or equal to 80.5. Remember, this is the continuity correction that you always need to make when approximating a discrete random variable with a continuous one. Okay, the next step is to standardize, and to do that we subtract the mean and divide by the standard deviation. So this is the same as the probability that z is greater than or equal to 80.5, take away 85, divided by the square root of 85. And that's about the same as the chance that z is greater than or equal to minus 0.49. Now that's not something we'd be able to look up in the tables, but if you think about it, that's the same as the probability that z is less than or equal to positive 0.49 because of the symmetry in the standard normal distribution curve. So we need to look up the probability that z is less than or equal to 0.49, and if you look in the tables, we'll see that that's 0.6879. And so that's the answer to the question. The probability that we're looking for is 0.6879. Okay, so that's the normal approximation to the Poisson distribution. Here's the main thing that you need to remember. When lambda is more than 10, then the Poisson distribution with parameter lambda, where lambda is the expected number of events, can be approximated by the normal distribution where the mean is lambda and the variance is also lambda. Okay, that's the end of my video about using the normal distribution to approximate the Poisson distribution. I hope you found it helpful. Thank you very much for watching.