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In order to apply anomaly detection, we're going to need to use the Gaussian distribution, which is also called the normal distribution. When you hear me say either Gaussian distribution or normal distribution, they mean exactly the same thing. If you've heard the bell-shaped distribution, that also refers to the same thing. But if you haven't heard of the bell-shaped distribution, that's fine too. But let's take a look at what is the Gaussian or the normal distribution. Say x is a number, and if x is a random number, sometimes called the random variable, x can take on random values. If the probability of x is given by a Gaussian or normal distribution with mean parameter Mu, and with variance Sigma squared. What that means is that the probability of x looks like a curve that goes like this. The center or the middle of the curve is given by the mean Mu, and the standard deviation or the width of this curve is given by that variance parameter Sigma. Technically, Sigma is called the standard deviation and the square of Sigma or Sigma squared is called the variance of the distribution. This curve here shows what is p of x or the probability of x. If you've heard of the bell-shaped curve, this is that bell-shaped curve because a lot of classic bells say in towers were shaped like this with the bell clapper hanging down here, and so the shape of this curve is vaguely reminiscent of the shape of the large bells that you will still find in some old buildings today. Better looking than my hand-drawn one. There's a picture of the Liberty Bell. Indeed, the Liberty Bell's shape on top is vaguely bell-shaped curve. If you're wondering what does p of x really means? Here's one way to interpret it. It means that if you were to get, say, 100 numbers drawn from this probability distribution, and you were to plot a histogram of these 100 numbers drawn from this distribution, you might get a histogram that looks like this. It looks vaguely bell-shaped. What this curve on the left indicates is not if you have just 100 examples or 1,000 or a million or a billion. But if you had a practically infinite number of examples, and you were to draw a histogram of this practically infinite number of examples with a very fine histogram bin. Then you end up with essentially this bell-shaped curve here on the left. The formula for p of x is given by this expression; p of x equals 1 over square root 2 Pi. Pi here is that 3.14159 or it's about 22 over 7. Ratio of a circle's diameter circumference times Sigma times e to the negative x minus Mu, the mean parameter squared divided by 2 Sigma squared. For any given value of Mu and Sigma, if you were to plot this function as a function of x, you get this type of bell-shaped curve that is centered at Mu, and with the width of this bell-shaped curve being determined by the parameter Sigma. Now let's look at a few examples of how changing Mu and Sigma will affect the Gaussian distribution. First, let me set Mu equals 0 and Sigma equals 1. Here's my plot of the Gaussian distribution with mean 0, Mu equals 0, and standard deviation Sigma equals 1. You notice that this distribution is centered at zero and that is the standard deviation Sigma is equal to 1. Now, let's reduce the standard deviation Sigma to 0.5. If you plot the Gaussian distribution with Mu equals 0 and Sigma equals 0.5, it now it looks like this. For practical anomaly detection applications, you usually have a lot of different features. You've now seen how the Gaussian distribution works. If x is a single number, this corresponds to if, say you had just one feature for your anomaly detection problem. But for practical anomaly detection applications, you will have many features, two or three or some even larger number n of features. Let's take what you saw for a single Gaussian and use it to build a more sophisticated anomaly detection algorithm. They can handle multiple features. Let's go do that in the next video.

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