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Overview of Random Variables and Probability Distributions

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This overview explains random variables and their associated probability distributions. A random variable assigns a numerical value determined by chance to each outcome in a procedure. Probability distributions can be represented through graphs, tables, or formulas that illustrate the likelihood of each possible value of a random variable. The text distinguishes between discrete random variables, which have finite or countable values, and continuous random variables, which can assume an infinite range of values without gaps. Additional concepts such as mean, variance, and standard deviation are also discussed.

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Overview of Random Variables and Probability Distributions

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    1. Sections 4.1 and 4.2 Overview Random Variables

    2. PROBABILITY DISTRIBUTIONS

    3. COMBINING DESCRIPTIVE METHODS AND PROBABILITIES

    4. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure. A probability distribution is a graph, table, or formula that gives the probability for each value of a random variable.

    5. EXAMPLES

    6. SAMPLE SPACE FOR ROLLING A PAIR OF DICE

    8. DISCRETE AND CONTINUOUS RANDOM VARIABLES A discrete random variable has either a finite number of values or a countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they can be associated with a counting process. A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.

    9. EXAMPLES Let x be the number of cars that travel through McDonald’s drive-through in the next hour. Let x be the speed of the next car that passes a state trooper. Let x be the number of As earned in a section of statistics with 15 students enrolled.

    10. PROBABILITY HISTORGRAM

    11. EXAMPLES

    12. REQUIREMENTS FOR A PROBABILITY DISTRIBUTION

    13. EXAMPLES

    14. MEAN, VARIANCE, AND STANDARD DEVIATION

    15. Enter values for random variable in L1. Enter the probabilities for the random variables in L2. Run “1-VarStat L1, L2” The mean will be . The standard deviation will be sx. To get the variance, square sx. FINDIND MEAN, VARIANCE, AND STANDARD DEVIATION WITH TI-83/84 CALCULATOR

    16. ROUND-OFF RULE FOR µ, s, AND s2

    17. MINIMUM AND MAXIMUM USUAL VALUES

    18. EXAMPLE

    19. IDENTIFYING UNUSUAL RESULTS USING PROBABILITIES Rare Event Rule: If, under a given assumption the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. Unusually High: x successes among n trials is an unusually high number of successes if P(x or more) is very small (such as 0.05 or less). Unusually Low: x successes among n trials is an unusually low number of successes if P(x or fewer) is very small (such as 0.05 or less).

    20. EXAMPLE

    21. EXPECTED VALUE

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