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PROBABILITY AND STATISTICS
CHAPTER 4 NOTES
DISCRETE PROBABILITY DISTRIBUTIONS
I. INTRODUCTION TO RANDOM VARIABLES AND PROBABILITY DISTRIBTIONS
A. Random Variable x.
1. x represents a
outcome which could be a
or a
resulting from a statistical experiment.
a) For example: Take a multiple choice test with 20 questions. Your raw score will be the
number of correct responses.
1) x = number of correct responses.
2) We can
the values for x: 0, 1, 2, 3,….20.
3) A
for the random variable is x = 18.
b) For example:
the mileage you drive between your home and your job.
1) x = number of miles you drive
2) We
list ALL of the possible responses here, since all decimal
numbers are possible.
3) A possible value for the random variable is x = 26.835 miles.
2. The term
is used because the particular value it takes in one trial of an
experiment occurs by chance.
B. Types of random variables.
1.
.
a) Values can be
b) Usually determined by a
process.
c) A discrete random variable takes on only
values.
d) The only values are usually
, however, an exception would be a shoe
size, which could include half sizes. However, a shoe size could not be 7.23.
2.
a) Possible values cannot be put in a list since there are
many.
b) Usually determined by a
process.
c)
measures in an interval (including decimals and fractions) are candidates for
replacement.
d)
values are skipped.
C. Probability distribution for a
random variable
1.
each value x that may occur in the experiment and assign the probability of its
occurrence P(x).
a) P(x) = (number of times x has occurred) divided by (total number of times the experiment is
performed)
b) The
of all the simple probabilities is 1. ∑ P(x) = 1
c) The
of a discrete probability distribution (also called the mean)
is denoted μ.
1) μ = ∑xP(x)
a. To calculate μ, make a table.
1. List the
in a vertical column I.
2. List the corresponding
in vertical column II.
3. In vertical column III, calculate the product of each value of x times its
probability. (Multiply column I by column II)
4. The expected value of x is the sum of the products listed in column III.
d) The
of a probability distribution is denoted σ.
1) Since we use the same table in which we have calculated the expected value, columns I
and II already are formed and have x and P(x) entries.
2) Subtract the mean μ from each value of x, thus computing the deviations (x – μ).
Form a vertical column IV for these deviations.
3) Square each deviation and form vertical column V for (x – μ)2, the deviations squared.
4) Multiply each deviation squared by the corresponding probability from column II. Put
these products into vertical column VI.
5) Add the entries in column VI.
6) Find the square root of the sum of column VI.
ROUND OFF RULE!!
Always round the mean, variance, and standard deviation to
DECIMAL place than you have in your
.
D. Probability distribution of a
random variable.
1. Since all the possible values of the random variable x can not be listed for a continuous variable, it
would be
to form a table such as we did for a discrete random variable.
2. With a continuous random variable, we deal with the probability that x occurs within an interval of
numbers.
At this time we will not be required to evaluate these probabilities, but will deal with them in later
chapters.
3. To compute the expected value μ and the standard deviation σ in a continuous probability
distribution, it would be necessary to use mathematics beyond the level of this course. Therefore,
when they are discussed in later chapters, we will be given the results of these computations.
Section 4-1 Examples
Identify each of the following as either a discrete or continuous random variable.
A. The number of people who are in a car
B. The number of miles you drive in one week.
C. Weight of a box of cereal.
D. The number of boxes of cereal you buy in a year.
E. Length of time you spend for lunch.
F. The number of patients on a psychiatric ward in one day.
G. The volume of blood which is transfused during an operation.
In a personality inventory test for passive-aggressive traits, the possible scores are 1= extremely passive; 2 =
moderately passive; 3= neither; 4 = moderately aggressive; and 5 = extremely aggressive. The test was
administered to a group of 110 people and the results were as follows:
X (score)
1
2
3
4
5
F (Freq.)
19
23
32
26
10
Construct a probability distribution table and compute the expected value (the mean) and the standard
deviation. Use the histogram
to graph the probability distribution.
Complete the following table: Remember that P(x) = f/110, since n = Σf = 110.
x
1
2
3
4
5
Σ
f
19
23
32
26
10
110
P(x)
x * P(x)
x-µ
(x - µ)2
(x - µ)2 * P(x)
Bonus Question: In a game you roll 2 dice. You win $5 if the sum of the dice is 3, 7, or 11. You lose $3 if the sum is
4, 5, 8, 9 or 10. For any other roll, you do not win or lose anything. What is the expected value for
this game? Is this a fair game?
SECTION 4-2
I. BINOMIAL PROBABILITIES
In the previous chapter, we looked at statistical experiments and computed the probabilities of specified events. We will now
examine a particular type of statistical experiment called a
.
A. Characteristics of a binomial experiment.
1. The same action is repeated
a) Conditions for repetition must be
.
b) One trial must be
of all others. (the results of one trial cannot affect another.)
c) The number of trials is
.
2. A binomial experiment must have
outcomes
a)
(defined in the problem)
b)
(all outcomes that do not qualify as successes.)
3. On an individual trial
a) P (success) =
b) P (failure) =
c) q = 1 – p (They are
events)
4. The number of trials that are successful is denoted r where r ≤ n.
B. The basic problem is to find the probability of getting exactly r successes out of n trials
1. In a binomial experiment we use r as the
.
a) r counts the number of
.
b) r is a discrete random variable
c) For each experiment, the possible values of r can be listed, r = 0, 1, 2, 3, … n.
The values range from r = 0 (no successful outcomes) to r = n (all outcomes are successful).
2. The probability of getting exactly 0 successes (all failures) is denoted
, the probability of getting exactly 1
success is denoted
, and so on.
a) Since the categories of no successes, exactly one success, exactly two successes, etc. are
events, we can add their probabilities to answer a question using the
combination.
b) The sum of all probabilities =
. P(0) + P(1) + P(2) + P(3) + ….. P(n) =
.
C. Methods of determining the probability of r successes out of n trials.
1. Formula
2. Table of Binomial Probabilities found in Appendix II of your text.
3. The TI-84 can solve a binomial distribution problem for us, so that’s the way we are going to go.
a. Press 2nd VARS (DISTR).
b. This takes you to a screen that lists 16 different types of statistical distributions. We will be returning to this
screen A LOT!!
1) You are looking for the 11th and/or 12th type of distribution listed.
a) Pressing the Alpha and A keys takes you to the binompdf function.
1. binompdf gives you the probability that you will receive PRECISELY (exactly) r
successes.
b) Pressing the Alpha and B keys takes you to the binomcdf function.
1. binomcdf gives you the CUMULATIVE probability for up to r successes.
c) You can also just scroll through the list to find these distributions.
c. Once you have selected the type of distribution you want, either binompdf or binomcdf, you enter the
in
fashion.
1) The calculator screen will prompt you for trials, p, and x.
a) enter the values asked for, highlight Paste, and press Enter.
b. Press the Enter key again to receive your answer.
e. As an example, let’s go through Question Number 2 on Page 235.
1) n =
(number of questions)
p=
, or
(probability of guessing the correct answer on one question)
r=
for part a,
for part b, is greater than or equal to
for part c, and greater than or
equal to
for part d.
2) Let’s do part a.
a) This is asking us for the probability of getting PRECISELY 10 correct, so we use the
.
b) 2nd VARS A gives us binompdf(
c) Fill in the values 10,.2,10, highlight Paste and press Enter.
1. Your screen should have binompdf(10,.2,10) showing.
d) Press Enter. The screen will tell you that the probability of getting all ten questions correct is
1.024 x 10-7. That means that you have a 0.0000001024 chance of getting them all correct.
3) part b asks for the probability of getting none of them correct.
a) This is also asking for a precise probability, so we again go to binompdf.
b) This time, we enter 10,.2,0 into the calculator and get the answer of .1073741824. That means
that you have a 10.7% chance of getting a zero on that quiz.
4) part c asks for the probability of getting at least one right. This is a
probability, as we
need to add the probability of 1 to the probability of 2, all the way up to the probability of 10.
a) The smart way to do this is to realize that the
of “at least 1” is “0”. Since we have
already found P(0) to be .107, we can subtract that from 1 and get the answer of .893
5) part d asks for the cumulative probabilities for r = 5,6,7,8,9, and 10. The calculator can’t do that, but it
can find the probability of r = 0,1,2,3, and 4. This is the
of 5 and up, so we can
this from
to get our answer.
a) binomcdf(10,.2,4) gives the cumulative probabilities for r = 0,1,2,3, and 4. This number is .9672.
When this is subtracted from 1, we get .0328. So, you have about a 3% chance of getting at least
a 50 on that quiz. Maybe we should try studying!!
6) The calculator will
give you the probability of
or equal to the number
you enter for x. If you want greater than x, you need to use the
rule.
Section 4-2 Examples
For each of the following:
State whether it is a binomial experiment or not.
If it is a binomial experiment, describe “success” and “failure”.
Identify values for n, q, and the range of values for r.
U.S.A. Today reported on July 27, 1990, that 70% of the people questioned said that they watched less T.V. than they did a year ago, 22%
said they watch the same amount, and 8% said they watch more. Find the probability out of a randomly selected group of 5 that exactly 3
will say they watch less T.V. this year than last.
There are 20 M&M candies in a dish. 8 are brown, 3 are red, 5 are green and 4 are yellow. Two candies are picked from the dish at random.
What is the probability that both are red?
A ten question multiple choice test is given. Each question has four choices. You did not study and have no clue as to any of the answers, so
you have to randomly guess each answer. What is the probability you guess exactly 6 correctly (and pass)?
U.S.A. Today reported on July 27, 1990, that 70% of the people questioned said that they watched less T.V. than they did a year ago,
22% said they watch the same amount, and 8% said they watch more.
Find the probability out of a randomly selected group of 5 that exactly 3 will say they watch less T.V. this year than last.
Find the probability that between 2 and 4 (inclusive) people will say they watched less T.V. this year than last year.
Find the probability that at least one will say they watched less T.V. this year than last year.
SECTION 4-3
More Discrete Probability Distributions.
I.. The Geometric Distribution
We use the Geometric distribution when we want to know how many times we may have to try something to get a success.
A. A Geometric Distribution is a discrete probability distribution of a random variable x that satisfies the following conditions:
1. A trial is
until a
occurs.
a) You will take the English SOL test until you pass it, no matter how many tries it takes!!!
2. The repeated trials are
of each other.
3. The probability of success, p, is
for each trial (which is just another way of saying that they are
independent).
B. The formula for finding the probability of a geometric distribution is 𝑃(𝑥) = 𝑝(𝑞) 𝑥−1 , where q – 1 = p.
C. The good news is that the calculator will also find the probability of a geometric distribution.
1. 2nd VARS will again get us to the list of distributions. geometpdf and geometcdf are the last two on the list.
2. Enter the probability of success, the number of trials you are interested in, and press Enter.
II. The Poisson Distribution
We use the Poisson distribution when we want to know the probability that a
of occurrences takes place
within a
of
or
. (How likely is it that an employee will miss 15 days of work in a year?)
A. The Poisson Distribution is a discrete probability distribution of a random variable x that satisfies the following conditions:
1. The experiment consists of
the number of times x, an event, occurs in a
.
2. The probability of the event occurring is the
for each interval.
3. The number of occurrences in one interval is
of the number of occurrences in other intervals.
B. The Poisson distributions are also on the calculator, between the binom and geomet distributions.
1. Enter the average number of occurrences for the desired interval, and the number you are interested in.
Paste and Enter to find the probability.
SECTION 4-3 EXAMPLES
You are a telemarketer. From experience, you know that the probability that you will make a sale on any given call is 0.23.
Find the probability that your first sale on any given day will occur on the fourth or fifth call.
Find the probability that the first sale will occur before your fourth sales call?
Two thousand brown trout are introduced into a small lake. The lake has a volume of 20,000 cubic meters.
What is the probability that three brown trout are found in any given cubic meter of the lake?