Sinopsis
Probability as defined by Webster's dictionary is "the chance that a given event will occur". Examples that we are familiar with are the probability that it will rain the next day or the probability that you will win the lottery. In the first example, there are many factors that affect the weather—so many, in fact, that we cannot be certain that it will or will not rain the following day. Hence, as a predictive tool we usually assign a number between 0 and 1 (or between 0% and 100%) indicating our degree of certainty that the event, rain, will occur. If we say that there is a 30% chance of rain, we believe that if identical conditions prevail, then 3 times out of 10, rain will occur the next day. Alternatively, we believe that the relative frequency of rain is 3/10. Note that if the science of meteorology had accurate enough models, then it is conceivable that we could determine exactly whether rain would or would not occur. Or we could say that the probability is either 0 or 1. Unfortunately, we have not progressed that far. In the second example, winning the lottery, our chance of success, assuming a fair drawing, is just one out of the number of possible lottery number sequences. In this case, we are uncertain of the outcome, not because of the inaccuracy of our model, but because the experiment has been designed to produce uncertain results.
The common thread of these two examples is the presence of a random experiment, a set of outcomes, and the probabilities assigned to these outcomes. We will see later that these attributes are common to all probabilistic descriptions. In the lottery example, the experiment is the drawing, the outcomes are the lottery number sequences, and the probabilities assigned are 1/iV, where N = total number of lottery number sequences. Another common thread, which justifies the use of probabilistic methods, is the concept of statistical regularity.
Content
- Introduction
- Computer Simulation
- Basic Probability
- Conditional Probability
- Discrete Random Variables
- Expected Values for Discrete Random Variables
- Multiple Discrete Random Variables
- Conditional Probability Mass Functions
- Discrete iV-Dimensional Random Variables
- Continuous Random Variables
- Expected Values for Continuous Random Variables
- Multiple Continuous Random Variables
- Conditional Probability Density Functions
- Continuous AT-Dimensional Random Variables
- Probability and Moment Approximations Using Limit Theorems
- Basic Random Processes
- Wide Sense Stationary Random Processes
- Linear Systems and Wide Sense Stationary Random Processes
- Multiple Wide Sense Stationary Random Processes
- Gaussian Random Processes
- Poisson Random Processes
- Markov Chains
- Assorted Math Facts and Formulas
- Linear and Matrix Algebra
- Summary of Signals, Linear Transforms, and Linear Systems
