Transcript probability distribution

Discrete Probability
Distributions
GOALS
1.
2.
Define the terms probability distribution and
random variable.
Distinguish between discrete and
continuous probability distributions.
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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
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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
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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
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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 Distribution
Variance 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
   2  1.290  1.135
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Continuous Probability
Distributions
GOALS
1.
2.
3.
4.
5.
Understand the difference between
discrete and continuous distributions.
Compute the mean and the standard
deviation for a uniform distribution.
Compute probabilities by using the
uniform distribution.
List the characteristics of the normal
probability distribution.
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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The Uniform Distribution - Example
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?
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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 Distribution
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It is bell-shaped and has a single peak at the center of the
distribution.
The arithmetic mean, median, and mode are equal
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.
It is symmetrical about the mean.
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.
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
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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:
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Sampling Methods and
the Central Limit Theorem
GOALS
1.
2.
3.
Explain why a sample is the only feasible
way to learn about a population.
Describe methods to select a sample.
Explain the central limit theorem.
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Why Sample the Population?
1.
2.
3.
To contact the whole population would be
time consuming.
The cost of studying all the items in a
population may be prohibitive.
The physical impossibility of checking all
items in the.
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Probability Sampling
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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
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Simple Random Sample: A sample
formulated so that each item or person in the
population has the same chance of being
included.
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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
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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.
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For example, you choose a random start page and
take 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
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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
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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
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Stratified Random Sampling: A population is
first divided into subgroups, called strata, and
a sample is selected from each stratum.
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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
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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
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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
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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.
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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).
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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
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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.
x  x
x 
2

2
n
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Using the Sampling Distribution of the Sample Mean
(Sigma Known)
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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:
z
X 

n
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Using the Sampling Distribution of the Sample Mean
(Sigma Unknown)
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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:
X 
t
s n
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