 Welcome to our lecture on probability distributions. Let's talk about probability distributions. A probability distribution describes the likelihood of various outcomes for a, let's call it an experiment, really a stochastic process. For example, suppose you want to know what is the likelihood of getting five heads and five coin tosses. Okay, five heads is one of the outcomes. Of course, you can get zero heads, you can get one head. So the way you would determine that is you'd be using what's called a discrete probability distribution. The one we'll use is the binomial. You can actually get the probability of five heads, four heads, three heads. Notice these are discrete. You can't get 1.67 heads. Now sometimes we're working with continuous measurements. For example, you want to determine the probability that an American adult weighs between 200 and let's say 220 pounds. Or you want to know what is the probability you have to wait between 20 and 40 minutes for a bus. Now wait and time, as we know, those are continuous measurements that we need to work with a continuous probability distributions. The thing, the one way of knowing something is a probability distribution. If you look at all the outcomes, add them up, and it should add up to one. Because the sum of all the probabilities of every outcome is one. Let's start out with a simple example, the toss of a single die. A die is a six sided cube with a different number on each side. The number is one, two, three, four, five, six. And the outcome of the toss is whatever number is face up after you toss the die. So that's our random variable with random outcomes. And assuming the die is fair, not biased in any way, the outcomes are all equally likely. You're just as likely to get a one or two or three or four or five or six. And so each one has the probability of one out of six, probability of one sixth. And if you like, you can call it a uniform distribution. But right now, all we want to see is that we can create a distribution by listing all the possible outcomes of a particular stochastic process. And alongside of each outcome, the probability of getting that outcome. There are two types of probability distributions, depending on the type of the random variable that you're studying. The random variable, remember, is the variable whose value is determined by chance. It's the outcome of a stochastic process, of a random process. The random variable, as we saw in the previous slide, can be either a discrete random variable or it could be a continuous random variable. For a discrete random variable, we would have to use a discrete probability distribution. And for a continuous random variable, we would have to use a continuous probability distribution. The reason is we can't use one for the other. A discrete random variable can only take on very specified, distinct values. And a continuous random variable can take on continuous values, can take on any value within a continuous interval. So naturally the way we describe the probability distribution is going to be different for these two different types of random variables. There are three major discrete probability distributions. One we're going to learn in this course is the binomial. And from the word buy, you know there are only two outcomes, like defect and not defect. So sometimes companies will actually study how many defective parts are there in an electric car, for example. Now again, the binomial distribution can answer the question. Now another discrete probability distribution is the hyper geometric distribution. Now this one is especially important if you're going to be a card shark and you like to play poker. The two outcomes without replacement and you could for example determine what is the likelihood of getting four aces when you play poker. In any case, we won't be doing it in this course. The third one is the Poisson distribution, which probably won't be covered in this course, but you may see it in other courses. So what is the Poisson distribution? And you have a discrete random variable and a continuous interval. For example, the number of defects per square yard of material. We can also simply enumerate the possible outcomes and indicate the probability for each one. We're going to see an example in the next slide. The probabilities for all the listed outcomes must add up to one. That's what makes it a probability distribution. You could have a mathematical formula to represent the discrete probability distribution and to compute probabilities. Those are the ones that we talked about before. The very well known discrete probability distributions. But you could also construct a discrete probability distribution by listing all the possible discrete outcomes from this random process. And for each outcome, you would have a probability associated with it. And you see a little example on the side of the toss of a die, which you saw before, but now it's looking like a probability distribution. So if we look at the discrete probability distribution more generally, we can see that there are three major properties that apply no matter what you're looking at. All the probabilities taken together sums to one. If you're enumerating or if we could do it differently with a formula, but the sum of all possible probabilities of all possible outcomes has to be one. That's your universe. That's 100%. If you come out to something less than one, it means you forgot an outcome. If you come out to something more than one, that may be a bigger problem because it probably means that your outcomes are not mutually exclusive. Imagine in the example you see here that I have not only the outcomes listed, but the outcome, the die comes up odd, which would include one and three and five. So there's overlap there. That wouldn't be overlap with the individual outcomes that are listed. So the sum has to be one, not less than one, not more than one, but it has to be one. The probability of any particular outcome is greater than or equal to zero, and we know from one, from number one it has to be less than one, but we also know that from the rules of probability that we learned in the last lecture. So any probability has to be between zero and one. Not less than zero and not greater than one. And finally, when you substitute a value of the random variable into the function, assuming you're using a function, like the well-known distributions on the previous slide, then your function computes to the probability of getting that value. Another thing you can see if we had to give you another type of discrete probability distribution, other than what was listed on the previous slide, the probability distribution for the toss of a die is a uniform distribution. It's a uniform discrete probability distribution because all the outcomes are equally likely, and since there are six of them, every outcome has the same probability as everything else, one, six. Let's talk about the expected value. What is an expected? It's very much like a mu. It's a single average value that summarizes the probability distribution. The best example, and the easiest one perhaps, is what do you expect to get when you toss a die? Only six outcomes. You either get a one, a two, a three, a four, five, or a six. They all have the same probability, one, six, one, six, one, six, one, six, one, six. So if you toss a die in average, what will you get? Let's see it a different way. Let's say we have a class of 40 students. Each one tosses a die, and then we average out what they got. The expected value is 3.5. How do I know that? Well, just common sense. You'll probably get as many sixes as ones, and the average of six and one of course is three and a half. You'll get as many twos as fives. The average of two and five is three and a half. You'll get as many threes and fours, as many threes as fours, and the average of three and four is three and a half. On average, you'll get three and a half. Now of course, if you toss a die, you're not getting a three and a half, but this is what it averages out if you keep doing this over and over again. That's called the expected value. We don't really want to use intuition in order to figure out the expected value of a distribution. It's easy with a die, but with other distributions, it may not be that easy. We want to be able to use a formula that's called mathematical expectation in order to compute that one average value, that long-term average, over the whole distribution. Let's call our random variable x. x takes on values, and we know that when we construct the discrete probability distribution, every value is associated with a probability. If we take each value, multiply it by the probability, and then sum all those products up, we end up with the expected value for the distribution. You can see the formula there. The expected value, e of x, is the sum of x times p of x over all values of i, over all outcomes of this random variable. This is going to end up being a weighted average. When we saw the distribution of the die, it's actually a straight average, but that's because all the outcomes were equally likely. When they're not all equally likely, you end up with a weighted average, and that's your expectation. That's the mathematical expectation. That can be used as an average, and it actually computes to the parameter of the distribution, the parameter called mu, the mean. We're going to compute the expected value when you toss a die. Remember the formula is, we're using d now, the sum of the d, that's the outcome, times its probability. You take one times a six to get a six, two times a six is two six, three times a six is three six, four times a six is four six, five times a six is five six, and six times a six is six six. Now you have to sum that column. That's the last column there, and you get 21.6. The 21 divided by six, 21 divided by six is three and a half. This proves that the expected value when you toss a die is 21 over six, or 3.5. We've looked at the expected value of the mu in effect. Just like there's a mu, this should be a variance in the standard deviation. We have sigma squared, the population variance, and we have a formula, and you can just follow the steps and you can see that we ended up with a variance when you toss a die of 2.92, and the standard deviation is 1.71. The expected value is an important one concept, and now we're going to learn about expected monetary value mostly used in decision making and business. You want to know what is the value, a lottery, or in fact when you invest in a product, you know there's a certain probability you'll make so much money, or a different probability for making a different amount of money. We can always compute the expected value using the formula which we just learned, the sum of the x times the probability of x. So here's a game. Let's call it a lottery. What are you going to call it? It could be a business investment, and the three outcomes $300, $120, and zero. And suppose your subjective probability, you believe that a one-third chance, you'll make a $300. It doesn't have to be subjective, it could be an actual roulette wheel and it works out that there's a one-third chance of $300, a one-third chance of $120, or a one-third chance of $0. On average what do you think you'll be making each time? Remember, one-third of the time you'll make $300, one-third of the time you'll make $120, and one-third of the time make $0. So using the formula $300 times a third plus $120 times a third plus $0 times a third, you end up with an expected monetary value or expected value of $140. That means on average you're going to make $140 per game if you play this game for a long, long time. This is very important in business decision making, or if you're looking at the lotteries and the value of that lottery. We have another example of EMV. In this example, we're basing our decision on the toss of a coin. Of course, I don't think that's the way you're going to do it when you are the decision maker in business. But this is framed as a game, so you can think of it as a game, or you can think of it as a business decision. The coin is tossed. If the coin comes up heads, you win $100. If the coin comes up tails, you lose $50. What's the expected value of the game? Okay, why do we need to know what the expected value is? Well, maybe you have to pay to play. The answer is the expected value of the game is $25. How do you get expected value? The same way we got it before. The outcome times the probability, $100 times one-half is $50. Negative $50 times one-half is negative $25. You add those up and you get $25. Why $25? What does that mean? Remember, mathematical expectation which holds for expected monetary value is the expected long-term value of this game or this decision or this lottery. What that means is that the assumption here is that you're doing this game or making this same type of decision repeatedly under identical circumstances many, many, many, many times. Half of the time you'll win $100 approximately. Half of the time, approximately, you'll lose $50 on the toss of the coin. So that over the long term, if we take the winnings and we take the losings and we add them up and divide by the number of tosses, over the long term you can expect to make approximately $25 per toss. So if you had 100 tosses, 50 of them you'd say you won $100, 50 of them you lost $50. So over the 100 tosses you make $2500, divide by the number of tosses you have $25 per toss, per game played. So no matter what anyone tells you for this particular game, don't pay more than $25 per toss to play. Example three is very similar. Instead of tossing a coin we have probabilities for each outcome, for each of three outcomes. One, it's framed as a type of a game. One outcome is that you might lose $100 and there's a 1 quarter chance of that happening. You might be ahead, you might win $80, there's a 1 quarter chance of that happening. The rest of it, half of the time, you come out even. You don't lose, you don't gain. So the question is as always how much should you be willing to pay to play? And for that you have to figure out the expected monetary value for playing one game. And you see here it comes out to negative $5. How did that happen? Negative 100 times 1 quarter because that happens 1 quarter of the time. Plus 80 times 1 quarter, zero times anything is zero and when you add that up you end up with a loss of $5. How do you pay negative $5 for playing a game? I think I'll leave that up to you to discuss. This example might remind you of the lottery ticket and in fact that's what it represents. We'll look at a lottery ticket and you get one chance in $5 million of making a million dollars, the jackpot, four chances in $5 million of making $100,000 and the rest which would be $4,999,995 over $5 million of making zero. Remember all the problems have to add up to one. And somebody says well how much is this lottery ticket worth? Well if you do the formula, a million dollars times one over $5 million which is $0.20, $100,000 times four and $5 million that's $0.08, zero times anything is zero, add it up you get $0.28. Basically this lottery ticket should not be worth more than $0.28 now of course if they charge $0.28 to the lottery ticket they break even. So assuming you're buying this from the city or state they're going to charge you like $1 for this lottery ticket for the privilege and if you don't know statistics and you don't realize the expected value is a lot less than $1 you might be foolish enough to pay $1 but clearly according to mathematicians the expected value is only $0.28. Here we have another lottery, it's not that different from what we had before although the values work out a little differently, not much but a little. There's a $5 million payout the probability of that $5 million payout is how many zeros are there? $0.0001 it's probably very close to a typical lottery ticket which may be even worse. The usual lottery tickets are probably much worse than this. If you look at EMV, expected monetary value, the expected value of this lottery is $5. You shouldn't pay more than $5 for a ticket. Tickets will always be more even more than $9 because people hold out hope that they are going to be the one who wins. Another reason we don't use expected monetary value typically as an individual when we're buying a lottery ticket is that most of us don't do the same decision under identical circumstances many many many many times because that's what EMV represents, expected value represents that type of decision. When people pay $9 or $10 or $20 for a lottery ticket like this, they're not using EMV for their decision making. Same thing with slap machines in a casino. Any kind of gambling is the game is going to look something like this. Slap machines are much worse. They're loaded much worse than this. You know who uses EMV in gambling? The house. The lottery commission that sets up this lottery. The casino administration that sets up the probabilities for the payouts and the slap machines, they're using EMV. They know exactly what they're making over the long term for all of their decisions. What do people use if they're not using EMV and making decisions like this? You may learn this in another course. You're not going to learn it in this course, but just so you know it's something called utility theory. A continuous random variable, if you recall, is a random variable that can take on any value inside of an interval. If you have a variable like, let's say, weight of adult males, it's not going to go down to the negative numbers. It's not even going to go down to zero, but between some reasonable values on the low side and on the high side, like maybe 75 pounds to 700 pounds, right? The random variable can take on any value. It doesn't have to be a whole number. It's not specific, distinct values. You can't enumerate. You can't create a continuous probability distribution the same way you can a discrete probability distribution. You can't just list possible outcomes and say, well, you know, what probability is associated with this? We have to ask for probabilities inside of an interval. I can't ask a question like, what's the probability that an individual adult males weight will be exactly 95 pounds? I can't ask that question because what does 95 pounds mean? Is it 94.99999? Well, that's not 95. It's a continuous random variable. The probability that a continuous random variable takes on a particular value over all the infinite possible values it could take on, it's zero. The probability that a random variable, weight of an adult male, will be exactly 95 would be that one possible value divided by an infinite number of basically it's zero. So we can't do it that way anymore. We can't ask what's the probability that our random variable will take on any one particular value. We have to ask our questions our probability questions about intervals. What's the probability that our random variable will take on a value, just looking at the picture here, a value between zero and one. There we've got looking at the picture we've got a continuous random variable with zero at the middle, zero is clearly the mean. And we want to know what's the probability that this random variable takes on any outcome between zero and one. If we consider the entire area under the curve, that curve is the function of that random variable, of the probability of that random variable. If we consider the entire area under the curve as the sum total of all of the probabilities well that's one. Remember the sum of all the possible probabilities is going to be one. Another way to look at it is if we take the definite integral, the shaded area under the curve, sum it up and divide by the total area under the curve for the entire interval of the entire random variable from let's say in this case negative infinity to plus infinity, we get a proportion. We get a proportion of time that this random variable will take on values inside that shaded interval. Well that's great because if we could define a proportion we can consider it a probability. A proportion is the same as a probability is the same as a percentage in the case of looking at continuous probability distributions. From the previous slide remember that for any continuous probability distribution, and there are a lot of them, some we will study and some we will not study, the sum total of the area under the curve is always one. That's the universe. The universe is 100% of what you're looking at. What are we going to look at? Well definitely we're going to do a lot with the normal distribution. That's a continuous probability distribution and it's going to take up quite a bit of time in our course. A variation of the normal distribution, the standard normal as another one we're going to do a lot with. Another variation of the normal distribution, the student's t-distribution the chi-square distribution and the f-distribution are optional statistics. So depending on the what course you're taking you may or may not be doing it in this introductory statistics course this semester. And another one that will not be done in our course and it's a good distribution that's used a lot but it's not used in an introductory statistics course is the exponential distribution. The exponential distribution is also a continuous probability distribution and just to give you an example if you've ever seen a learning curve or a decay curve those are examples of exponential distribution. Thank you for attending our lecture. In this lecture we introduced the concept of a probability distribution for a random variable. If the random variable is a discrete random variable we need to look at a discrete probability distribution and we're going to go further in depth in one example of a discrete probability distribution, the binomial distribution. If the random variable is a continuous random variable we need a continuous probability distribution and we're going to go further in depth into mainly the normal distribution in that lecture and in further lectures where we look at variations of the normal distribution. Practice, practice, practice, do your homework, do every problem you can find. There are lots and lots of problems in various places on the website. Even when you did this lecture I hope you paused it at the appropriate time and did the problem first and then listened to the narration. The only way that you will do well is to do very, very well in this course and in fact not only do well but you'll find it easy is if you do lots and lots of problems. Practice, practice, practice.