Chapter 4 4 1 4 2 random variables
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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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Chapter 4 4 1 4 2 random variables

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


Warm up

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.


Random variable x

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.


Random variable x1

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?


Types of random variables

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


Discrete probability distributions

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


Discrete probability distributions cont

Discrete Probability Distributions (cont.)

  • The following table represents a probability distribution. What is the missing value?


Discrete probability distributions cont1

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


Mean and standard deviation of a probability distribution

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!


Mean and standard deviation of a probability distribution cont

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:


Expected value

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.


Expected value1

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?


Assignment

Assignment

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


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