The next probability distribution that I want to discuss is the normal distribution. And the normal distribution is given in this extract from the reference handbook over on the right, and is given by this expression. Function of x is one over sigma, squared of 2 pi, e to the minus one half, x minus mu, over sigma squared. Where in that equation sigma is the standard deviation of the population and mu is the mean value of the population. And the normal distribution is also called the Gaussian distribution. It's a symmetrical bell shaped curve. And it's probably one of the most important in all of probability and statistics. It describes many real world phenomena. We often express it in terms of the standardized or unit normal distribution, which is given by this expression. Function of x is one over square to two pi, e to the minus x squared over two. Where we note that this variable, z x minus mu over sigma, which you can think of as the number of standard deviations from the mean, also follows a normal distribution. And the shape of this curve if we plot it out, is the familiar bell-shaped curve which looks like this, and in this curve of course, now this is normalized we can think of this as x over sigma on the horizontal axis the number of standard deviations from the mean, and in between plus and minus one standard deviation. The area under that curve is approximately 68% of the total. Between plus and minus two standard deviations, this area here is approximately 95% of the total area. And finally, between plus and minus three standard deviations, is approximately 99.7% of the total area out of the curve. This is tabulated function, we won't normally need to calculate it. And this is an extract from the table given in the reference handbook. So the first column here is x and here we have the function of x, which is just the Bell-shaped curve here, the next column, capital x, capital f of x, is the area under the curve from minus infinity up to x, in other words, this area right here. The next column, r of x, is the area under the curve from x to infinity, in other words, this area. And the next column is twice that area, 2Rx. The last column, W of x, is the area under the curve from -x to +x. This column, this area here. So let's do an example on that. The mean rainfall in a city can be approximated as a Gaussian distribution. The mean rainfall is 15 inches, and the standard deviation is two inches. So the first question is, what is the probability that the rainfall in any year is between 18 and 20 inches? Which one of these. So here is our Gaussian standardized curve. Looks like this. Which is given by this equation. So what we want to do here is calculate our normalized variable z is x minus mu over sigma, and we're looking for the two points, which correspond to 18 inches of rainfall or 20 inches of rainfall. And what we want is the area under the curve between those two limits. This area right here. So we calculate the Z values. So the lower Z value Z1 is equal to 18 inches. The lower limit of the rainfall- the mean value which is 15. Divided by the standard deviation is two is equal to 1.5. So the lower limit here Z1 is 1.5. The upper limit Z2 is the upper limit of the rainfall 20 inches- the mean, over 2 which is equal to 2.5 so this is this area here. And the next step is to go to those tables such as the one on the previous slide, and look up the area. And the area, capital F, we can do it this way. So first all, the area F1 is this area, from minus infinity up to Z1, is .9332. Then we calculate the area or find the area from the table up to the upper limit which is this area function Z2 is equal to .9938. And then the area between the two, is just the difference between those two values which is .9938 minus .9332 Which is .0606, or rounding it off, is equal to 6% so the answer is A. And of course we could have found that area directly, by looking at the column of the area between those two limits would be another possibility. The next question, the probability that the rainfall in any one year is greater than 18 inches, is which of these alternatives? So in this case, we're looking for the probability that the area is greater than some value. In other words, the area to the right of the Z1 value, which is this area right here. So, again we can go to the table and the parameter here is R of Z1, so R evaluated at Z1, which is 1.5, is equal to .0668. Which is of course equal to one minus this number, one minus f of Z1, so, rounding that off, that is equal to 6.7% or, rounding off to a whole integer, is equal to 7%. Is the probability that the rainfall is greater than 18 inches in any one year. And this completes my preliminary discussion of the normal distribution.
