Stats 2035 Chapter 4 Notes

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Western University
Statistical Sciences
Statistical Sciences 2035
Histro Sendov

Chapter 4: Discrete Random Variables Two Types of Random Variables  Random variable: A variable that assumes numerical values that are determined by the outcomes of an experiment, where one and only one numerical value is assigned to each experimental outcome.  Discrete random variable: o A random variable that takes on a finite number or countable infinite number of values  Continuous random variable: o A random variable that assumes any numerical value in one or more interval on the real number line. Discrete Probability Distributions  Probability distribution (Of a random variable): A table, graph, or formula that gives the probability associated with each possible value that a random variable can assume. o Probability distribution of a discrete random variable x is denoted as p(x) o P(x) 0 for each value of x. o ∑ ( )  Mean of a discrete random variable: ∑ ( ) o The value expected to occur in the long run  Variance of a discrete random variable: ∑ ( ) ( ) o Measures the spread of the values of the random variable from their expected value.  Standard deviation of a discrete random variable: √ o Positive squared root of the variance. o Chebyshev’s theorem application: o For every integer k 1, p(x) falls in the interval [ ] 1 – 1/k Binomial Distribution  Binomial experiment: o The experiment consists of (n) identical trials. o Each trial results in a success or a failure. o The probability of a success on any trial is p, and the remains constant from trial to trail. This implies that the probability of failure, q, on any trial is 1- p, and remains constant from trial to trail. o The trials are independent (each result has no relation to each other)  Binomial random variable (x): The total number of successes in n trials of a binomial experiment  Binomial formula: ( ) ( ) ⁄ ( ) o The probability of getting x successes in n trials o = The probability of getting x successes in a particular combination o The number of ways to arrange x successes in n trials ⁄ ( )  Binomial table: Tables in which we can look up binomial probabilities 
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