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Chapter 4 4.1-4.2: Random VariablesPowerPoint Presentation

Chapter 4 4.1-4.2: Random Variables

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### 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

- 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 - 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 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

- 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

- 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

- All of the probabilities must be between 0 and 1.
- The sum of the probabilities must equal 1.
- ∑ P(x) = 1

Discrete Probability Distributions (cont.)

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

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:
- 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

- Calculate as you would for a weighted mean or frequency distribution:

Very important!

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

- 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 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

- pp. 190 # 2 – 14 Even, 18 – 22 Even

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