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Chapter 4 4.1-4.2: Random Variables

CHS Statistics. Chapter 4 4.1-4.2: Random Variables. Objective : Use experimental and theoretical distributions to make judgments about the likelihood of various outcomes in uncertain situations. Warm-Up. Decide if the following random variable x is discrete(D) or continuous(C).

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Chapter 4 4.1-4.2: Random Variables

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  1. CHS Statistics Chapter 44.1-4.2: Random Variables Objective: Use experimental and theoretical distributions to make judgments about the likelihood of various outcomes in uncertain situations

  2. Warm-Up • Decide if the following random variable x is discrete(D) or continuous(C). • X represents the number of eggs a hen lays in a day. • X represents the amount of milk a cow produces in one day. • X represents the measure of voltage for a smoke-detector battery. • X represents the number of patrons attending a rock concert.

  3. Random Variable X • Random variable - A variable, usually denoted as x, that has a single numerical value, determined by chance, for each outcome of a procedure. • Probability distribution – a graph, table, or formula that gives the probability for each value of the random variable.

  4. Random Variable X • A study consists of randomly selecting 14 newborn babies and counting the number of girls. If we assume that boys and girls are equally likely and we let x = the number of girls among 14 babies… • What is the random variable? • What are the possible values of the random variable (x)? • What is the probability distribution?

  5. Types of Random Variables • A discrete random variable has either a finite number of values or a countable number of values. • A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a ways that there are no gaps or interruptions. • Usually has units

  6. Discrete Probability Distributions • A Discrete probability distribution lists each possible random variable value with its corresponding probability. • Requirements for a Probability Distribution: • All of the probabilities must be between 0 and 1. • 0 ≤ P(x) ≤ 1 • The sum of the probabilities must equal 1. • ∑ P(x) = 1

  7. Discrete Probability Distributions (cont.) • The following table represents a probability distribution. What is the missing value?

  8. Discrete Probability Distributions (cont.) • Do the following tables represent discrete probability distributions? 1) 2) 3) 4) • 5) P(x) = x/5, where x can be 0,1,2,3 • 6) P(x) = x/3, where x can be 0,1,2

  9. Mean and Standard Deviation of a Probability Distribution • Mean: • Standard Deviation: • Calculator: • Calculate as you would for a weighted mean or frequency distribution: • Stat  Edit • L1 = x values • L2 = P(x) values • Stat  Calc • 1: Variable Stats L1, L2 Very important!

  10. Mean and Standard Deviation of a Probability Distribution (cont.) • Calculate the mean and standard deviation of the following probability distributions: 2) Let x represent the # dog per household: 1) Let x represent the # of games required to complete the World Series:

  11. Expected Value • The expected value of a discrete random variable represents the average value of the outcomes, thus is the same as the mean of the distribution.

  12. Expected Value • Consider the numbers game, often called “Pick Three” started many years ago by organized crime groups and now run legally by many governments. To play, you place a bet that the three-digit number of your choice will be the winning number selected. The typical winning payoff is 499 to 1, meaning for every $1 bet, you can expect to win $500. This leaves you with a net profit of $499. Suppose that you bet $1 on the number 327. What is your expected value of gain or loss? What does this mean?

  13. Assignment • pp. 190 # 2 – 14 Even, 18 – 22 Even

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