Ardavan Asef-Vaziri:	>> Let's solve a couple of normal distribution problem. Normal probability distribution is the most applied distribution in the real world, weight of people, test scores, life of a light ball, number of votes in an election, demand for the product, cost of a product, summation of random variables from other distributions. They all follow normal distribution. Normal distribution is symmetric. The left hand side is mirror image of the right hand side, and it is a bell shaped. Mean, median, and mode are equal to each other's. Normal distribution is defined by two parameters, average, Mu and standard deviation, Sigma. We use this notation to show variable X follows normal distribution with average of Mu and the standard deviation of Sigma. The mean can be any numerical value, negative, zero, or positive. It can be negative too. Standard deviation defines the width of the curve. The larger the standard deviation, the wider the curve. Coefficient of variations like any other distribution is standard deviation divided by average. Suppose we have a specific X from a specific normal distribution with a specific average, specific standard deviation, and we want to know what is the probability of the random variable to be less than or equal to this specific X value. Given that specific X, this area to the left of that X defines the Probability. So this is visual representation. This is the probability that we are looking for. Give me this X. This is my graphical solution, but sometimes graphical solution is not enough and we need to also provide numerical solution. That numerical probability is equal to Norm Dist X, Mu Sigma, and one. That is the formula used to show probability of a random variable to be less than equal to something. Now, given the probability if I have the probability and I'm looking to find X value. Give me the X value for which probability to the left is equal to something. In that case, when I have probability and I want X, I use this formula. If we have X and we are looking for probability, we use this formula, and if we have probability and we are looking for X, we use this formula, and now I go through a couple of examples. Demand for a product follows normal, average 1,000, standard deviation 150. What is the probability that the demand is less than or equal to 900. We have X, and we are looking for the probability. We use norm distribution function, Norm Dist, X 900, Mu 1,000, standard deviation 150 and the last component one because you want to compute probabilities. That's what you want. What is the probability that the demand is at least 1,100? That means we are looking for this probability. We find less than equal to 1,100 and then subtract it from one and that is what we need. Everything is like that one, 1,000, 150, one, 1,100, but we need to subtract it from one because this is this side, we are looking for this side. Cost of a product follows normal distribution with average of 1000 and 150. What is probability that the cost is 1,100-1,300? Quite straightforward. Here I have 1,100, here I have 1,300, and I want a probability between them. This is probability of less than equal to 1,300. We put it into a simple equation that we have already learned and we compute it. This is probability of less than or equal to 1,100, again, we put it into the same formula. The only difference is here, I have 1,300, here I have 1,100. I can compute that one too, and then I need to take this part out of this part to compute this part. I simply subtract it, and that is the result. Score of students in a test follows normal distribution with average of 800, standard deviation 120. What is the probability that the score is 760-920? Exactly the same as the previous part. We are looking for this area, which is 760 and 920. For 920, we find the area that is this area, and then we need to find this area and take it out. That is 760, we compute it, and then we subtract. Average weight of red deer follows normal distribution with average of 500 pounds and standard deviation of 75 pounds. What is the probability that a red deer weights between these two numbers? Exactly the same thing. We are looking for this probability. We find it for the larger number, and then we find it for the smaller number. Those are two probabilities. The first one, this probability, the second one is this probability. We take this one out of this one, and that is what we are looking for. Demand for a product follows normal distribution, average 1,000 standard deviation 150. The bottom 15% of demand is less than what value? This is 15%, and we want to look for this value. The lowest 15% of demand is smaller than what value? Norm Inverse, 15%. These two come from over there, and that is the value. How much inventory do we need to make sure that probabality of demand exceeding our inventory does not exceed 10%. We wanted this area to be 10% and if that area is 10%, this area is 90%. Therefore, it is quite straightforward. We are looking for a point for which the probability to the left is 90%. This is 90%, 1 - 0.1, this was 0.1, and these are average and standard deviation, and we find the value. Thank you very much for paying attention.