Chap5-8x

Random Variables and Probability Distributions Chapters 5-8

McGraw-Hill/Irwin

Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Learning Objectives Chapter 5 1. Define probability. 2. Define Random Variable. 3. Understand discrete random variable and continuous random variable. Chapter 6 Identify the characteristics of a probability distribution. Chapter 7 1. List the characteristics of the normal probability distribution. 2. Convert a normal distribution to the standard normal distribution. 3. Find the probability for a normally distributed random variable. 4. Find the Z-value.

5-2

Learning Objectives Chapter 8 1 Explain why a sample is often the only feasible way to learn something about a population. 4 Describe the sampling distribution of the sample mean. 5 Understand the central limit theorem. 6 Define the standard error of the mean. 7 Apply the central limit theorem to find probabilities.

8-3

A Survey of Probability Concepts Chapter 5

McGraw-Hill/Irwin

Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Random Variables (r.v) RANDOM VARIABLE: A quantity resulting from an experiment that, by chance, can assume different values. Random variable is usually denoted by X. Discrete r.v.: A random variable that can assume only certain clearly separated values. 1. The outcome when throwing a die of six sides 2. The number of students in a class. 3. The number of cars entering a carwash in a hour. Continuous r.v.: can assume an infinite number of values within a given range. 1. The weight of each student in this class. 2. The temperature outside as you are reading this book. 3. The amount of money earned by each of the more than 750 players currently on Major League Baseball team rosters.

Probability PROBABILITY: A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur. A probability is the relative possibility that a random variable assumes certain values.

Notations: For discrete r.v.: P(X = x) or P(x) For continuous r.v.: P(X > x)

Discrete Probability Distributions Chapter 6

McGraw-Hill/Irwin

Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Probability Distribution PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome. Experiment:

Outcome

P(X=x)

Toss a die of six sides. The outcome of the experiment is a r.v, denoted by X. The possible results are: 1, 2, 3, 4, 5, and 6.

1

1/6

2

1/6

3

1/6

4

1/6

5

1/6

What is the probability distribution for each of the outcome?

6

1/6

Total

1

Characteristics of a Probability Distribution 1.The probability of a particular outcome is between 0 and 1 inclusive. 2. The outcomes are mutually exclusive events. 3. The list is exhaustive. So the sum of the probabilities of the various events is equal to 1. 4. For continuous distributions, P(X = x) = 0

Continuous Probability Distributions Chapter 7

McGraw-Hill/Irwin

Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Normal Probability Distribution 1. 2.

3.

4.

It is bell-shaped and symmetrical about the mean. The area under the curve denotes probability. The total area under the curve is 1.00. The location of a normal distribution curve is determined by the mean , the shape of the curve is determined by the standard deviation,σ. For a r.v. that follows normal distribution with mean  and s.d. σ, we denote X~(, σ).

mean 

The Family of Normal Distribution

The Standard Normal Probability Distribution 

 



The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is also called the z distribution. A r.v. that follows standard normal distribution is denoted by Z. Converting a normally distributed r.v., X to a standard normally distributed r.v., Z: subtract X by its mean and divide by its s.d. Z score

Probabilities: Areas Under Curve P(Z > 1.80)= P(Z < -1.80) = .0359; P(Z < −1.52) = .0643;

=NORMSDIST(-1.8) =NORMSDIST(-1.52)

P(−1.30 < Z < 2.10) = P(Z