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STAT 2103 Class Topics Chapter 4 thru Chapter6(1)(3) (1)

STAT 2103 Class Topics Chapter 4 thru Chapter6(1)(3) (1) -...

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STAT 2103 Class Topics Chapter 4 A random variable is a function or rule that assigns a number to each outcome of an experiment. Alternatively, the value of a random variable is a numerical event. Two types of random variables: Discrete Random Variable – one that takes on a countable number of values – E.g. values on the roll of dice: 2, 3, 4, …, 12 Continuous Random Variable – one whose values are not discrete , not countable – E.g. time (30.1 minutes? 30.10000001 minutes?) Analogy: Integers are Discrete, while Real Numbers are Continuous A probability distribution is a table, formula, or graph that describes the values of a random variable and the probability associated with these values. Since we’re describing a random variable (which can be discrete or continuous) we have two types of probability distributions: – Discrete Probability Distribution, and – Continuous Probability Distribution An upper-case letter will represent the name of the random variable, usually X. Its lower-case counterpart will represent the value of the random variable. The probability that the random variable X will equal x is: P(X = x) or more simply P(x) 1
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The probabilities of the values of a discrete random variable may be derived by means of probability tools such as tree diagrams or by applying one of the definitions of probability, so long as these two conditions apply: 1. for all x 2. for all x Distribution of a Discrete Random Variable Population Mean Population Variance Population Standard Deviation Example x P(x) 0 .48 0*(.48)=0 0^2*(.48)=0*(.48)=0 1 .35 1*(.35)=.35 1^2*(.35)=1*(.35)=.35 2 .08 2*(.08)=.16 2^2*(.08)=4*(.08)=.32 3 .05 3*(.05)=.15 3^2*(.05)=9*(.05)=.45 4 .04 4*(.04)=.16 4^2*(.04)=16*(.04)=.64 Total 1.00 =.82 =1.76 2
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= E(X) = = 0(.48) + 1(.35) + 2(.08) + 3(.05) + 4(.04) = .82 = = 1.0429 Binomial Probability Model for Bernoulli Trials A binomial probability model describes the number successes in a specified number trials. It takes two parameters to specify this model: the number of trials n and the probability of success. The binomial distribution is the probability distribution that results from doing a “ binomial experiment ”. Binomial experiments have the following properties: Fixed number of trials, represented as n . Each trial has two possible outcomes, a “success” and a “failure”. P(success)=p (and thus: P(failure)=(1–p), for all trials. The trials are independent , which means that the outcome of one trial does not affect the outcomes of any other trials. Success and Failure are just labels for a binomial experiment, there is no value judgment implied. For example a coin flip will result in either heads or tails. If we define “heads” as success then necessarily “tails” is considered a failure (inasmuch as we attempting to have the coin lands heads up). Other binomial experiment notions: An election candidate wins or loses An employee is male or female The random variable of a binomial experiment is defined as the number of successes in the n trials, and is called the binomial random variable .
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