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STATS 2B03 (25)
Chapter 4

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Aaron Childs

Stats 2B03: Statistical Methods for Science Chapter 4: Probability Distributions 4.2 Probability Distributions of Discrete Variables - The probability distribution of a discrete random variable is a table, graph, formula, or other device used to specify all possible values of a discrete random variable along with their respective probabilities - - ∑ for all x - Cumulative distributions: may be obtained by successively adding the probabilities. Graph of a cumulative probability distribution is called an ogive - Mean and variance of discrete probability distributions: ∑ ∑ ∑ , where p(x) is the relative frequency of a given random variable X. the standard deviation is simply the positive square root of the variance 4.3 The Binomial Distribution - When a random process or experiment, called a trial, can result in only one of two mutually exclusive outcomes, the trial is called a Bernoulli trial - The Bernoulli process: a sequence of Bernoulli trials forms a Bernoulli process under the following conditions:  Each trial results in one of two possible, mutually exclusive, outcomes. One of the possible outcomes is denoted (arbitrarily) as a success, and the other is denoted as a failure  The probability of a success, denoted by p, remains constant from trial to trial. The probability of a failure, 1 – p, is denoted by q  The trials are independent; that is, the outcome of any particular trial is not affected by the outcome of any other trial - Large sample procedure – use of combinations: a combination of n objects taken x at a time is an unordered subset of c of the n objects, n - Binomial table: gives the probability that X is less than or equal to some specified value. Give the cumulative probabilities from x=0 up through some specified positive number of successes - Using table B when p>.5: | | - The binomial parameters: n and p are parameters in the sense that they are sufficient to specify a binomial distribution 4.4 The Poisson Distribution - If x is the number of occurrences of some random even in an interval of time or space, the probability that x will occur is given by - The poisson process: binomial distribution results from a set of assumptions about an underlying process yielding a set of numerical observations:  The occurrences of the events are independent. The occurrence of an event in an interval of space or time has no effect on the probability of a second occurrence of the event in the same, or any other, interval.  Theoretically, an infinite number of occurrences of the event must be possible in the interval.  The probability of the single occurrence of the event in a given interval is proportional to the length of the interval  In any infinitesimally small portion of the interval, the probability of more th
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