The last discrete random variable that we will discuss is the so-called geometric random variable. It shows up in the context of the following experiment. We have a coin and we toss it infinitely many times and independently. And at each coin toss we have a fixed probability of heads, which is some given number, …
Probability – L05.7 Geometric Random Variables Read More »
The next random variable that we will discuss is the binomial random variable. It is one that is already familiar to us in most respects. It is associated with the experiment of taking a coin and tossing it n times independently. And at each toss, there is a probability, p, of obtaining heads. So the …
Discrete Uniform. In this segment and the next two, we will introduce a few useful random variables that show up in many applications– discrete uniform random variables, binomial random variables, and geometric random variables So let’s start with a discrete uniform. A discrete uniform random variable is one that has a PMF of this form. …
We now want to introduce some examples of random variables, and we will start with the simplest conceivable random variable– a random variable that takes the values of 0 or 1, with certain given probabilities. Such a random variable is called a Bernoulli random variable. And the distribution of this random variable is determined by …
Probability – L05.4 Bernoulli & Indicator Random Variables Read More »
A random variable can take different numerical values depending on the outcome of the experiment. Some outcomes are more likely than others, and similarly some of the possible numerical values of a random variable will be more likely than others. We restrict ourselves to discrete random variables, and we will describe these relative likelihoods in …
Probability – L05.3 Probability Mass Functions Read More »
We will now define the notion of a random variable. Very loosely speaking, a random variable is a numerical quantity that takes random values. But what does this mean? We want to be a little more precise and I’m going to introduce the idea through an example. Suppose that our sample space is a set …
Probability – L05.2 Definition of Random Variables Read More »
In this lecture, we introduce the notion of a random variable. A random variable is, loosely speaking, a numerical quantity whose value is determined by the outcome of a probabilistic experiment. The weight of a randomly selected student is one example. After giving a general definition, we will focus exclusively on discrete random variables. These …
In this segment, we will discuss the multinomial model and the multinomial probabilities, which are a nice generalization of the binomial probabilities. The setting is as follows. We are dealing with balls and the balls come into different colors. There are r possible different colors. We pick a ball at random, and when we do …
We will now apply our multinomial formula for counting the number of partitions to solve the following probability problem. We have a standard 52-card deck, which we deal to four persons. Each person gets 13 cards as, for example, in bridge. What is the probability that each person gets exactly one ace? Well, before we …
We now come to our last major class of counting problems. We will count the number of ways that a given set can be partitioned into pieces of given sizes. We start with a set that consists of n different elements. And we have r persons. We want to give n1 items to the first …