Discrete distributions

- [Instructor] In this course, you'll review four main types of probability distributions. A probability distribution is a mathematical function that shows all possible outcomes and their associated probabilities for an experiment. It essentially represents the probabilities of random events occurring in the denoted sample space. For example, the random variable X can denote the outcome of a fair coin toss, which is the experiment. The probability distribution of X would take the probability of 0.5 for the value of heads and 0.5 for the value of tails. The four probability distributions you will review in this course include discrete, continuous, cumulative, and joint. Discrete and continuous distributions are the base for probability distributions, and cumulative and joint distributions are special cases that come from both of these. Let's explore discrete probability distributions first. A discrete probability distribution is represented by discrete random variables that contain a finite or accountable number of values. This distribution denotes the probability that a discrete random variable has a certain value. Remember the previous roll of a die example. The discrete probability distribution will represent the outcomes of one through six and their associated probabilities of one over six. There are three main properties that discrete probability distribution needs to satisfy. First, the probability of each outcome in the discrete random variable in the sample space needs to be between the values of zero and one. Second, the set of possible outcomes must be countable. And like mentioned prior, the sum of all the probabilities for the random variable in the sample space must equal exactly one. With these three properties satisfied, you can continue on with your discrete probability distribution. Probability distributions are represented by what is called probability functions. A probability function is a mathematical function that denotes the probabilities for the possible outcomes of a random variable. In this course, probability functions will be denoted as f with parentheses and a little x inside of it. For discrete probability distributions, they're represented by probability mass functions. The probability mass function, also known as a PMF, is a function that gives the probabilities of a discrete variable, which is a large X, being equal to their associated outcomes, a little x. This function can be represented by the following equation where you have your probability mass function equal to the probability that your random variable X is equal to an outcome little x. Probability mass functions can be represented by an equation or in the form of a table depending on the type of distribution being used. Probability mass functions have two main properties. This includes how the probability mass function f of x needs to be greater than zero for all x in the sample space and zero otherwise. This means each outcome denoted in the sample space needs to have a probability greater than zero with every other number not in the sample space containing a probability of zero since they are not possible outcomes. The other property is represented by this equation, where this means the sum of all the probabilities of all the possible outcomes in the sample space of the experiment must be equal to one to make it complete. Let's take a moment to look at an example of a discrete probability distribution for an experiment. Let's have the experiment be tossing two fair coins and the discrete random variable X representing how many tails can occur. The sample space of this experiment contains four outcomes, including two heads, a heads and a tail, a tail and a heads, and finally two tails. In this case, the H represents the heads and T denotes the tails. In this scenario, it is possible to have zero, one, or two tails occur between the two coins being tossed. For the discrete probability distribution, you'll need to calculate the probability of zero, one and two tails occurring from the coin tosses. From this experiment, you can calculate the probabilities by dividing the number of times the outcome is possible by the total number of possible outcomes. For example, there's one probability of there being zero tails in four outcomes. This makes the probability of zero tails to be one fourth or 0.25. Note that you can also calculate these in a table or histogram since there are very few outcomes to work with. Now you see that discrete probability distributions are important for many experiments in understanding their potential outcomes and associated probabilities. In a later chapter, you'll dive deeper into exploring different discrete probability distributions and their associated probability mass functions. The discrete distributions you'll explore in this course include discrete uniform, Bernoulli, binomial, negative binomial, geometric, hyper geometric, and finally, Poisson. Let's switch over to introducing continuous probability distributions.