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C4: DISCRETE RANDOM VARIABLES CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics. 2007 Longin Jan Latecki Discrete Random Variables Discrete random variables are obtained by counting and have sample spaces which Are countable. The values that represent each outcome are usually integers. Random variables are denoted by capital letters. X: is the number of times that we flip a coin until H comes up The possible outcomes are denoted by lower case letters: a=1, a=2, a=3... Probability Mass Function The probability mass function (pmf) of a discrete random variable maps each possible outcome in the sample space to it's corresponding probability. The sum of the probabilities of all possible outcomes will always be equal to 1. Probability Distribution Function The distribution function of a random variable X, also referred to as the cumulative distribution function (CDF) yields the probability that X will take a value less than or equal to a. Hence, the value of F(a) is equal to the sum of all probabilities of outcomes less than or equal to a: F (a) P( X ai ) P( X a ) P( X a ) ai a If we are given a CDF, we can get the pmf with the following formula: P( X a ) P( X a ) P( X a ) P( X a ) P( X a ) F ( a ) F ( a ) for some sufficiently small ε>0. Graphs of pmf and CDF Cumulative Distribution Function Probability Distribution Function F (a) P( X ai ) P( X a ) P( X a ) ai a Bernoulli Distribution The Bernoulli distribution is used to model an experiment with only two outcomes, success and failure. The parameter p is the chance for success. An example is flipping a coin, where “heads” may be success and “tails” may be failure. Binomial Distribution The Binomial Distribution represents multiple Bernoulli trials. The parameter n is the number of trials, and the parameter p is the probability of success as in the Bernoulli distribution. P(X=k) is the probability of k successful outcomes in n trials. Geometric Distribution A geometric distribution gives information about the probability of success after k attempts. The parameter p is the probability of success on the kth try. This means that all previous k-1 tries failed An example of this would be finding the probability that you will hit a bullseye with a dart on your kth toss.