This module is the complementary teacher's guide for the "Discrete Random Variables" chapter of the Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.
This chapter introduces
expected value (long term average) and four of the
common discrete random variables (binomial, geometric, hypergeometric, and Poisson). The authors cover expected value and two of the discrete random variables (binomial and Poisson). Depending on your background, you may want to cover the binomial (usually required) together with none or some of the other discrete random variables
Random variables
Explain
random variable (assigns numerical values to the outcomes of a statistical experiment). Upper case letters denote random variables. Example: Let
= the number of cars in your household. (The phrase "the number of" tells you that
takes on discrete values.)
takes on the values 0, 1, 2, 3, ...
The probability distribution function
A
probability distribution function (pdf) is best shown with an example: A controversial drug is given to two patients. Let X = the number of patients cured.
A pdf is easiest to understand in a table.
or
0
1
2
Each probability is between 0 and 1.
The previous example can be used as an example of
expected value or long term average (
). Make a third column labeled
. Calculate the three values and add them. The result,
, is the expected number of patients who are cured if the drug is administered many times to two patients.
The binomial is a special discrete pdf or
pattern . A binomial experiment consists of counting the number of successes in one or more Bernoulli trials. (A Bernoulli trial has only two possible outcomes, success or failure. In every Bernoulli trial, the probability of a success (or failure) remains the same.)
John comes to his stat class and discovers he must take a true-false quiz . There are 20 questions on the quiz. John has not attended class recently and must guess randomly at the questions. Let
= the number of questions John answers correctly out of 20 questions.
takes on the values 0, 1, 2, 3, ..., 20.
. John's guessing at the answers is a binomial experiment.
Notation:
~
where the number of trials,
, is 20 and the probability of a success,
, on any trial is 0.5.
Students can find the mean (
), and the standard deviation (
square root of
) either by hand or with technology. (
is the probability of a failure.) Have students help you fill in the blanks and answer the questions:
Draw the graph. (horizontal axis is the number of successes; vertical is the probability of 0 successes, 1 success, 2 successes, ..., 20 successes. Draw vertical lines or boxes.
What is the probability that John gets 15 questions correct?
More than 15 questions correct?
At least 15 questions correct?
A
geometric experiment takes place when at least one Bernoulli trial is performed and all are failures except the last one which is the only success. Example: Liz likes to play darts. The probability that she hits the bull's eye (success) on any throw is 85%. (Liz is good!) Liz throws darts at the bull's eye
until she hits it. Let
= the number of times Liz throws the dart at the bull's eye until she hits it. Have students help you fill in the blanks:
Fill in the blanks.
~ _______ (
~
where
probability of a success= 0.85)
Draw the graph. (Number of throws until the first success versus probability)
4. What is the probability that Liz hits the bull's eye for the first time on the third throw? That it takes more than three throws for Liz to hit the bull's eye for the first time? That it takes at least three throws?
takes on the values _______.
_______. In words,
is _______
The geometric equation
Hypergeometric distribution
The
hypergeometric distribution is characterized by choosing a sample without replacement from two distinct groups. One of the two groups is what is of interest in the sample. Some lotteries are based on the hypergeometric distribution.
click to edit note
Suppose a shipment of 20 tape recorders contains 5 defectives. An inspector randomly chooses 8 of the tape recorders to inspect. He is interested in the number of defectives in the sample of 8. Have the class answer questions similar to those for the binomial and the geometric.
~
where
= size of the group of interest,
= size of the other group, and
= size of the sample.
Poisson distribution
The Poisson distribution is concerned with the number of times an event takes place in a certain interval. It is used in the field of reliability. The Poisson approximates the binomial when n is "large" (say, more than 100) and p is "small" (say, less than 0.1).
Suppose the average number of accidents that occur in a week at a particularly busy intersection is one. The interval is one week. The average is one accident. Let
= the number of accidents that occur in a one week period at the intersection. Have the students help fill in the blanks and answer the questions:
_______
(
where
)
What values does
take on?
What is the probability that at most one accident occurs in a week?
The poisson distribution formula
The parameter for the Poisson is the mean,
. Some books and calculators use the Greek letter,
(lambda) as the mean. The equation for the Poisson is:
Assign practice
Have the students complete the
portion of the practice that is appropriate for what you have covered in class. Expected Value, Binomial, and Poisson are dealt with
Practice 1 ,
Practice 2 , and
Practice 3 .
Practice 4 is based on the Geometric Distribution, while
Practice 5 is focused on reviewing the Hypergeometric Distribution.
Calculator instructions
If you are using the TI-83/TI-84 series, there are probability functions for the binomial, Poisson, and geometric. Each has a pdf and a cdf (for example binompdf and binomcdf).These functions are located in 2nd DISTR. If you use, say,
binompdf(n,p) , you will get the table of probabilities for 0, 1, 2, ...,
. If you use
binompdf(n,p) , you will get the probability of
. If you use
binomcdf(n, p, x) , you will get the cumulative probability
.
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
