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

CHS Statistics

Decide if the following random variable x is discrete(D) or

continuous(C). 1) X represents the number of eggs a hen lays in a day.

2) X represents the amount of milk a cow produces in one day.

3) X represents the measure of voltage for a smoke-detector battery.

4) X represents the number of patrons attending a rock concert.

Warm-Up

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?

Random Variable XProbabilities of

Girlsx (Girls) P(x)

0 01 0.0012 0.0063 0.0224 0.0615 0.1226 0.1837 0.2098 0.1839 0.122

10 0.06111 0.02212 0.00613 0.00114 0

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

Types of Random Variables

A Discrete probability distribution lists each possible

random variable value with its corresponding probability.

Requirements for a Probability Distribution:1. All of the probabilities must be between 0 and 1.

0 ≤ P(x) ≤ 1

2. The sum of the probabilities must equal 1.

∑ P(x) = 1

Discrete Probability Distributions

The following table represents a probability

distribution. What is the missing value?

Discrete Probability Distributions (cont.)

x 1 2 3 4 5

P(x) 0.16 0.22 0.28 0.2  

Do the following tables represent discrete probability

distributions?1) 2) 3)

4)

Discrete Probability Distributions (cont.)

x P(x) 0 0.2162 0.4323 0.2884 0.064

x P(x)5 0.286 0.217 0.438 0.15

x P(x)1 1/22 1/43 5/44 -1

x P(x)1 .092 0.363 0.494 0.06

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:

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

Mean and Standard Deviation of a Probability

Distribution

Very important!

Calculate the mean and standard deviation of the

following probability distributions:

Mean and Standard Deviation of a Probability

Distribution (cont.)

1) Let x represent the # of games required to complete the World Series:

x P(x)

4 0.480

5 0.253

6 0.217

7 0.410

2) Let x represent the # dogs per household:

X = # of Dogs

Households

0 14911 4252 1683 48

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?

Expected Value

Event x P(x)

Win  

Lose  

pp. 190 # 2 – 14 Even, 18 – 22 Even

Assignment