# Review of Probablility and Statistics

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REVIEW OF PROBABILITY AND STATISTICS

Risk and Return Risk can be generally defined as the uncertainty of outcomes. It is best explained in terms of probability, which traces its roots to problems of fair distribution. In fact, in the Middle Ages the word probability meant an “option certified by authority.” The question of justice led to notions of equivalence between expectations. And work on expectations set the stage for probability theory. Probability traces its roots to the work of Girolamo Cardano, an Italian who was also an inveterate gambler. In 1565 Cardano published a treatise on gambling, Liber de Ludo Alae, which was the first serious effort at developing the principles of probability. Probability theory took another leap when a French nobleman posed a gambling problem to Blaise Pascal in 1654. He wanted to know how to allocate equitably profits in a game that was interrupted. In the course of developing answers to this problem, Pascal laid out the foundations for probability theory. Probability of the outcome Imagine an experiment where the outcome is not predictable. Cardano and Pascal defined probability distributions, which describe the number of times a particular value can occur in an imaginary experiment. Notation: p(Xi) – probability of Xi Intuitively, probability is a proportion of times that any particular outcome is observed out of total number of times that an experiment is repeated. Example: Coin toss. There are two possible outcomes: either a coin comes Heads up, or Tails up. The probability of getting a Head is ½. Random Variables and Probability Distributions Discrete Random Variables A random variable that can take on at most a countable number of possible values is called discrete. For a discrete random variable X, we define the probability mass function p(a) of X by pa   PX  a Probability mass function tells us the probability that random variable X is equal to a. Example 1: Consider for instance a gambler with a pair of dice....