Discrete (specific) probability laws Flashcards
A list of a variable’s values also with the probability of their occurrence is…?
A probability distribution
What is a probability density function (PDF)?
It is a function that describes the probability density at each point in a sample space.
Probability laws are?
Functions that define probability laws
What are the major types of distributions (of probability)?
- Empirical 2. Theoretical 3. Discrete 4. Continuous
Empirical are
values of varibale are collected through empirical observation
Theoretical are
values of variable are derived from mathematical function
Discrete are
Values of a variable that are countable
Continuous are
Values of variable exist along a continuum the discrimination between discrete and continuous can sometimes be arbitrary
How to we hope to apply distributions (of probability)?
By knowing how closely a frequency distribution from a sample (empirical distribution) matches a theoretical probability distribution from some corresponding population.
What are the two types of probability?
Theoretical probability and empirical probability
Describe theoretical probability
It is deductive (a priori – which means with prior knowledge) Looks into the future possibilities # of favor outcomes divided by total # of outcomes
What is an example of a theoretical probability?
the probability of a face card in a random (normal 52 card deck)? We already know the distribution of cards and all the outcomes therefore based on this knowledge (a priori) we are able to come up with a theoretical probability.
What is empirical probability?
Inductive (a posterior) relative frequency of past events # of times an event occurred divided by the total # of trials
What is an example of empirical probability?
probability of rain on August day in Albuquerque? We look at the historical data (a posterior) to determine the events that occurred and therefore compute a probability of future events occuring.
What are five important theoretical probability distributions for behavioral research?
- Discrete Binomial 2. Continuous normal, t, x2, F
What is the granddaddy of all distributions?
The normal distribution, more or less most distributions are related or rely on the normal distribution.
What is a formula that gives the probability of some sample statistic having a specific value?
A probability law
What is another way to think of a probability law?
A specific method for selecting entities in a population given some characteristic.
How do we define specific probability laws?
a random sample of n observations is taken from a population containing N units, each of which belongs tone and only one of K categories, the categories being, in effect the possible values that an observation may have.
What are two key questions we have about probability laws when they are defined?
How is sampling done? what is the nature of the categories in the population?
There can be two different types of sampling methods
- Depletable – sampling without replacement and N is finite 2. Undepletable - sampling with replacement OR N is infinite
What are different types of population categories?
- each unit belongs to only one category (therefore the categories are mutually exclusive) 2. each unit must belong to a category (therefore the categories are exhaustive) If you have 1. and 2. then you have a partition Finally, the number of categories in a population 2 = dichotomy 3+ = multichotomy
Given this framework what are the 4 distinct probability laws?
What is hypergeometric probability law?
a random sample of n observations is to be drawn without replacement from a finite population (depletable).
The population contains n units each of which belongs to one or the other of two mutually exclusive and exhaustive categories. (dichotomy of units)
How do statisticians typically call this dichotomy (with hypergeometric probability?
As successes and failures
This is an arbitrary distinction
Successes are usually called “R”
Failures are expressed as “N-R”
How is the hypergeometric probability law expressed?
What is this an example of?
A hypergeometric probability type of question.
How can we generalize the hypergemoetric law so that it is now multivariate?
This means there are more than two distinct categories (mutlichotomy)
What does this formula express?
Hypergeometric multivariate formula.
“At least” is special in proabilities, why?
It means “x or more”
What means “X or less” in word problems for probability?
“no more than X”
When word problems involve “at least” OR “ no more than” how do we obtain an overall probability?
One must add up the probabilities of each occurence.
An undepletable probability law is called?
Hint: it is the most famous probability law in all of statistics
The binomial probability law
What are the signficant attributes of the bionmial probability law?
- the outcomes of the individual observations are independent of each other.
- every unit belongs to one or the other of two mutually exclusive and exhaustive categories
- population is a dichotomy
- the sampled population is INFINITE
How are successes and failures expressed in the binomial probability law?
The proportion of units belonging to one category are labled as successes = “p”
The other proportion are labeled as failures “1-p”
What probability law is being expressed here?
The binomial probability law
What kind of probability law can be used to solve this equation?
The binomial probability law.
A generalization of the binomial probability law to the cae where the sample population contains more than two kinds of units is called what?
Multinomial probability law
A random sample of n observations is to be drawn either from an infinite population or with replacements from a finite population. And the outcomes are independent of each other
What law should we use?.
Either the binomial or multinomial (depending on the number of categories).
How is the multinomial probability law expressed?
What probability law would we use to solve this word problem?
Multinomial probability law
If I want to find the probability that a sample of n
observations will have to be taken in order to
obtain a predetermined number r of successes, which probability law should I use?
–this an infinite population
The negative bionmial probability law
“How many trials will it take before we get N successes”
How does one determine the probability that the rth success will occur on the nth observation (given an infinite population)?
The negative binomial probability law
What law is expressed here?
The negative binomial probability law
Write out the negative binomial probability law.
Write out the binomial probability law.
Write out the hypergeometric probability law
Write out the hypergeometric multivariate probability law
Write out the multinomial probability law
What kind of probabilty law would we use to solve this word problem?
We need a special case of the negative bionmial when the r=1, what is this called?
Geometric probability law
How is the geometric probability law applied?
Under the same conditions as the negative binomial probability law and therefore the binomial probability law
Write out the geometric probability law
Which probability law is expressed here?
Geometric probability law
What type of probability law solves this word problem?
Geometric probability law
IF we have N pairs of matched observations, we can ask whetehr the distribution ofone set of numbers is different in magnitutde from the otehr set of numbers.
What type of probability law is used?
The sign test which is a simple application of the binomial probabibilty law.
For the sign test, what do we assign as the default probability?
0.5 for success or failures, this implies that there is no greater than a equal chance for change and the distribution of variables will be normal about some mean. Therefore a deviation from this mean will be greater than chance regardless of the direction.
