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# Chapter 4 4.1-4.2: Random Variables PowerPoint PPT Presentation

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).

Chapter 4 4.1-4.2: Random Variables

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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

• 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

• 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

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