chapter 5: the binomial probability distribution and related topics section 1: introduction to...

Chapter 5Chapter 5: The Binomial : The Binomial Probability Distribution and Probability Distribution and

Related TopicsRelated Topics

Section 1Section 1: Introduction to : Introduction to Random Variables and Probability Random Variables and Probability

DistributionsDistributions

The quantitative variable x is a random The quantitative variable x is a random variable because the value that x takes on variable because the value that x takes on in a given experiment is a chance or in a given experiment is a chance or random outcome.random outcome.

Continuous Random VariableContinuous Random Variable – has an – has an infinite number of possible values that is infinite number of possible values that is not countable; usually measures the not countable; usually measures the amount of somethingamount of something

Discrete Random VariableDiscrete Random Variable - has either a - has either a finite number of possible values or a finite number of possible values or a countable number of possible values; countable number of possible values; usually counts somethingusually counts something

ExamplesExamples

1.1. The number of A’s earned in a math The number of A’s earned in a math class of 15 students class of 15 students

(Discrete)(Discrete)

2. The number of cars that travel through 2. The number of cars that travel through McDonald’s drive thru in the next hour McDonald’s drive thru in the next hour (Discrete)(Discrete)

3.3. The speed of the next car that passes a The speed of the next car that passes a state trooper state trooper

(Continuous)(Continuous)

probability distributionprobability distribution – an assignment of – an assignment of probabilities to specific values of the probabilities to specific values of the random variablerandom variable

Probability DistributionProbability Distribution

xx P(x)P(x)

00 ¼ ¼

11 ½ ½

22 ¼ ¼

Properties of Discrete Probability DistributionsProperties of Discrete Probability Distributions 0 0 << P(x) P(x) << 1 1 The sum of the values of P(x) for each distinct The sum of the values of P(x) for each distinct

value of x is 1.value of x is 1.

∑ ∑P(x) = 1P(x) = 1

xx P(x)P(x)

11 0.200.20

22 0.250.25

33 0.100.10

44 0.140.14

55 0.310.31

xx P(x)P(x)

11 0.200.20

22 0.350.35

33 0.120.12

44 0.400.40

55 -0.06-0.06

Which of the following is a probability distribution?

xx P(x)P(x)

11 0.200.20

22 0.250.25

33 0.100.10

44 0.140.14

55 0.490.49

A. B. C.

Determine the required value of the missing Determine the required value of the missing probability in order to make the distribution a probability in order to make the distribution a

discrete probability distribution.discrete probability distribution.

A. A. B.B.xx P(x)P(x)

33 0.400.40

44

55 0.100.10

66 0.200.20

xx P(x)P(x)

00 0.300.30

11 0.2150.215

22

33 0.200.20

44 0.150.15

55 0.050.05

Given a discrete Given a discrete random variable x random variable x with probability with probability distribution P(x), the distribution P(x), the meanmean of the random of the random variable x is defined variable x is defined as as

= ∑x ∙ P(x)= ∑x ∙ P(x)

= ∑ (x - )2∙P(x)= ∑ (x - )2∙P(x)

Given a discrete Given a discrete random variable x random variable x with probability with probability distribution P(x), the distribution P(x), the variancevariance of the of the random variable x is random variable x is defined asdefined as

2

*The standard deviation is the square root

of the variance.

ExampleExample

For the following probability distribution, find For the following probability distribution, find the mean, variance, and standard deviation.the mean, variance, and standard deviation.

xx P(x)P(x)

11 4/104/10

22 4/104/10

33 1/101/10