Discrete Probability Distributions

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1. Define the terms probability distribution and random variable.

2. Distinguish between discrete and continuous probability distributions.

GOALS

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What is a Probability Distribution?

Experiment: Toss a coin three times. Observe the number of heads. The possible results are: zero heads, one head, two heads, and three heads. What is the probability distribution for the number of heads?

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Probability Distribution of Number of Heads Observed in 3 Tosses of a Coin

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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. The sum of the probabilities of the various events is equal to 1.

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Random Variables

Random variable - a quantity resulting from an experiment that, by chance, can assume different values.

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Types of Random Variables

Discrete Random Variable can assume only certain clearly separated values. It is usually the result of counting something

Continuous Random Variable can assume an infinite number of values within a given range. It is usually the result of some type of measurement

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Discrete Random Variables - Examples The number of students in a class. The number of children in a family. The number of cars entering a carwash in a hour. Number of home mortgages approved by Coastal

Federal Bank last week.

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Continuous Random Variables - Examples

The length of each song on the latest Tim McGraw album.

The weight of each student in this class. The temperature outside as you are reading this

book. The amount of money earned by each of the more

than 750 players currently on Major League Baseball team rosters.

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The Mean of a Probability Distribution

Mean•The mean is a typical value used to represent the central location of a probability distribution.•The mean of a probability distribution is also referred to as its expected value.

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The Variance, and Standard Deviation of a Discrete Probability DistributionVariance and Standard Deviation• Measure the amount of spread in a distribution• The computational steps are:

1. Subtract the mean from each value, and square this difference.2. Multiply each squared difference by its probability.3. Sum the resulting products to arrive at the variance.

The standard deviation is found by taking the positive square root of the variance.

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Mean, Variance, and Standard Deviation of a Discrete Probability Distribution - Example

John Ragsdale sells new cars for Pelican Ford. John usually sells the largest number of cars on Saturday. He has developed the following probability distribution for the number of cars he expects to sell on a particular Saturday.

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Mean of a Discrete Probability Distribution - Example

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Variance and Standard Deviation of a Discrete Probability Distribution - Example

135.1290.12

Continuous Probability Distributions

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GOALS

1. Understand the difference between discrete and continuous distributions.

2. Compute the mean and the standard deviation for a uniform distribution.

3. Compute probabilities by using the uniform distribution.

4. List the characteristics of the normal probability distribution.

5. Define and calculate z values.

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The Family of Uniform Distributions

The uniform probability distribution is perhaps the simplest distribution for a continuous random variable.

This distribution is rectangular in shape and is defined by minimum and maximum values.

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The Uniform Distribution – Mean and Standard Deviation

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Southwest Arizona State University provides bus service to students while they are on campus. A bus arrives at the North Main Street and College Drive stop every 30 minutes between 6 A.M. and 11 P.M. during weekdays. Students arrive at the bus stop at random times. The time that a student waits is uniformly distributed from 0 to 30 minutes.

1. Draw a graph of this distribution.

2. How long will a student “typically” have to wait for a bus? In other words what is the mean waiting time? What is the standard deviation of the waiting times?

3. What is the probability a student will wait more than 25 minutes?

4. What is the probability a student will wait between 10 and 20 minutes?

The Uniform Distribution - Example

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The Uniform Distribution - Example

1. Draw a graph of this distribution.

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The Uniform Distribution - Example

2. Show that the area of this distribution is 1.00

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The Uniform Distribution - Example

3. How long will a student “typically” have to wait for a bus? In other words what is the mean waiting time? What is the standard deviation of the waiting times?

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The Uniform Distribution - Example

4. What is the probability a student will wait more than 25 minutes?

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The Uniform Distribution - Example

5. What is the probability a student will wait between 10 and 20 minutes?

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Characteristics of a Normal Probability Distribution1. It is bell-shaped and has a single peak at the center of the distribution.

2. The arithmetic mean, median, and mode are equal

3. The total area under the curve is 1.00; half the area under the normal curve is to the right of this center point and the other half to the left of it.

4. It is symmetrical about the mean.

5. It is asymptotic: The curve gets closer and closer to the X-axis but never actually touches it. To put it another way, the tails of the curve extend indefinitely in both directions.

6. The location of a normal distribution is determined by the mean(), the dispersion or spread of the distribution is determined by the standard deviation (σ) .

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The Normal Distribution - Graphically

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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 z-value is the signed distance between a selected value, designated X, and the

population mean , divided by the population standard deviation, σ. The formula is:

Sampling Methods Sampling Methods and and the Central Limit the Central Limit TheoremTheorem

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GOALS

1. Explain why a sample is the only feasible way to learn about a population.

2. Describe methods to select a sample.

3. Explain the central limit theorem.

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Why Sample the Population?

1. To contact the whole population would be time consuming.

2. The cost of studying all the items in a population may be prohibitive.

3. The physical impossibility of checking all items in the.

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Probability Sampling

A probability sample is a sample selected such that each item or person in the population being studied has a known likelihood of being included in the sample.

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Methods of Probability Sampling

Simple Random Sample: A sample formulated so that each item or person in the population has the same chance of being included. For example, let's say you were surveying first-time parents

about their attitudes toward mandatory seat belt laws. You might expect that their status as new parents might lead to similar concerns about safety. On campus, those who share a major might also have similar interests and values; we might expect psychology majors to share concerns about access to mental health services on campus.

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Methods of Probability Sampling

Systematic Random Sampling: The items or individuals of the population are arranged in some order. A random starting point is selected and then every kth member of the population is selected for the sample. For example, you choose a random start page and t

ake every 45th name in the directory until you have the desired sample size. Its major advantage is that it is much less cumbersome to use than the procedures outlined for simple random sampling.

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Systematic Random Sampling

If a systematic sample of 500 students were to be carried out in a university with an enrolled population of 10,000, the sampling interval would be:

I = N/n = 10,000/500 =20

Note: if I is not a whole number, then it is rounded to the nearest whole number.

All students would be assigned sequential numbers. The starting point would be chosen by selecting a random number between 1 and 20. If this number was 9, then the 9th student on the list of students would be selected along with every following 20th student. The sample of students would be those corresponding to student numbers 9, 29, 49, 69, ........ 9929, 9949, 9969 and 9989.

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Methods of Probability Sampling

Stratified Random Sampling: A population is first divided into subgroups, called strata, and a sample is selected from each stratum. For example, you are interested in product

preference between men and women. So, you divide your sample into male and female members and randomly select equal numbers within each subgroup (or "stratum"). With this technique, you are guaranteed to have enough of each subgroup for meaningful analysis.

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Methods of Probability Sampling

Cluster Sampling: A population is first divided into primary units then samples are selected from the primary units.

- Suppose an organization wishes to find out which sports school students are participating in across Bangkok. It would be too costly and take too long to survey every student, or even some students from every school. Instead, 100 schools are randomly selected from all over Bangkok.

These schools are considered to be clusters. Then, every student in these 100 schools is surveyed. In effect, students in the sample of 100 schools represent all students in Bangkok.

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Sampling Distribution of the Sample Means The sampling distribution of the

sample mean is a probability distribution consisting of all possible sample means of a given sample size selected from a population.

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Central Limit Theorem

When you throw a die ten times, you rarely get ones only. The usual result is approximately same amount of all numbers between one and six. Of course, sometimes you may get a five sixes, for example, but certainly not often.

If you sum the results of these ten throws, what you get is likely to be closer to 30-40 than the maximum, 60 (all sixes) or on the other hand, the minimum, 10 (all ones).

The reason for this is that you can get the middle values in many more different ways than the extremes. Example: when throwing two dice: 1+6 = 2+5 = 3+4 = 7, but only 1+1 = 2 and only 6+6 = 12.

That is: even though you get any of the six numbers equally likely when throwing one die, the extremes are less probable than middle values in sums of several dice.

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Central Limit Theorem

The central limit theorem explains why many distributions tend to be close to the normal distribution

The key ingredient is that the random variable being observed should be the sum or mean of many independent identically distributed random variables.

If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples.

The mean of the sampling distribution is equal to μ and the variance equal to σ2/n.

nxx

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x

40

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Using the Sampling Distribution of the Sample Mean (Sigma Known)

If a population follows the normal distribution, the sampling distribution of the sample mean will also follow the normal distribution.

To determine the probability a sample mean falls within a particular region, use:

n

Xz

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If the population does not follow the normal distribution but n ≥ 30, the sample means will follow the normal distribution.

To determine the probability a sample mean falls within a particular region, use:

ns

Xt

Using the Sampling Distribution of the Sample Mean (Sigma Unknown)