Discrete Distributions Flashcards
What is a distribution?
the possible values a variable can take and how frequently they occur
What is the probability function?
p(y)
What are probabilities or probability distributions?
measure the likelihood of an outcome
We define distributions with which two characteristics
- Mean & 2. Variance
What character do we use to denote mean?
population: μ - mu
sample: x-bar
What character do we use to denote variance?
population: σ2 - sigma squared
sample: s2 - s squared
What character do we use to denote std deviation?
population: σ - sigma
sample - s
What units are std deviation measured in?
The same units as the mean
What units is variance measured in?
mean units squared
What is a distribution?
A collection of all the possible values a variable can take and how frequently they occur in the sample space.
A function which assigns a probability to each value a variable can take.
The average value of the elements in the data set.
The likelihood of an event occurring.
A collection of all the possible values a variable can take and how frequently they occur in the sample space.
What is the difference between sample data and population data?
Sample data represents the entire data we have, while population data is only some part of it.
The terms sample data and population data are interchangeable and mean a part of the data.
The terms sample data and population data are interchangeable and mean the entire data set.
Sample data represents some part of the data, while population data is the same as all the data.
Sample data represents some part of the data, while population data is the same as all the data.
Which of the following is expressed with the letter s?
Population mean.
Population variance.
Population standard deviation.
Sample mean.
Sample variance.
Sample standard deviation.
Sample standard deviation.
What are the Types of Probability Distributions?
- Discrete Distributions - finite number of outcomes - examples: rolling a die or picking a playing card
- Continuous distributions - infinitely many outcomes - example: Time and distance
What is the notation for distributions?
X ~ N (μ, σ2)
Variable - Tilde - Type - Characteristics in parenthesis (usually mean, variance)
What are some characteristics of Discrete Distributions?
- all outcomes are equally likely - called Equiprobable
They follow a uniform distribution - finitely many distinct outcomes
- can express the entire probability distribution with either a table, graph or formula
- need to ensure every unique outcome has a probability assigned to it
In Probability we are often more interested in the likelihood of an interval than an individual value.
intervals - add up all the values that fall within that range
What is a Bernoulli Distribution?
- Events with only 1 trial and 2 possible outcomes
- Any event with two outcomes can be transformed into a Bernoulli event
– We use ‘Bern’ to describe it: Bern(p) - examples: coin flip, quiz with T or F question, Vote D or R
- usually denote the higher value with p - we assign a value of 1
- usually denote the lower value with 1- p
- The variance of Bernoulli events would always = p(1-p)
What is a Binomial Distribution?
- The outcomes per iteration are two
- Many iterations
- a sequence of identical Bernoulli events
- Notation: B(n,p)
What is the Poisson Distribution?
- Test how unusual an event frequency is for a given interval
- Denoted as Po(lamda)
- deals with the FREQUENCY with which an event occurs within a specific interval
- instead of the probability of an event, it requires knowing how often it occurs for a specific period of time or distance
- graph always starts from zero
example: a firefly lights up 3 times in 10 seconds. What is the likely hood of it lighting up 8 times in 20 sec
What is Euler’s number? Napier’s Constant?
e ~ 2.72
What are characteristics of Continuous Distributions?
The probability distribution graph would be a curve as opposed to unconnected individual bars
What is the Normal Distribution?
outcomes often observed in nature
What is the Chi-Squared? (Kai)
Asymmetric
only consists of non-negative values
always starts with zero on the left
What is the Exponential distribution?
used when dealing with events that are rapidly changing early on
online news example - more relevant when the news is fresh, as times goes on they die off
What is the Logistic distribution?
useful in forecast analysis
useful for determining a cut-off point for a successful outcome
example:
how much of an in-game advantage is necessary to predict victory for either team
predictions would never reach true certainty
What are the two main types of distributions based on the type of data we have? Multiple Choice
Distinct and Continuous.
Discrete and Continuous.
Finite and Infinite.
Discrete and Infinite.
Discrete and Continuous.
Why do we group distributions into types?
We don’t group distributions into types.
Because they share certain features.
Because they have different numbers of possible values.
None of the above.
Because they share certain features.
How do we compute the probability of an interval for a discrete distribution?
We can’t.
We add up the values for of the individual outcomes in the interval.
We add up the probabilities of each individual outcome in the interval occurring.
We express the probability distribution in a table and add up the probabilities of the two end points of the interval.
We add up the probabilities of each individual outcome in the interval occurring.
Which of the following is true for any variable Y following a discrete distribution? ** We use “<=” and “>=” to express “less than or equal to” and “greater than or equal to” respectively.
P(Y=y) = P(Y < y)
P(Y=y) > P(Y<=y)
P(Y < y) = P(Y <= y+1)
P(Y<=y) = P(Y < y+1)
P(Y<=y) = P(Y < y+1)
What are some characteristics of a Uniform Distribution?
- U(a,b) we use the letter U to define it, followed by the range of value in the dataset
- all outcomes have equal probability
- the expected value provides us with no real information because all outcomes have the same probability
- Each outcome is equally likely - both the mean and the variance is uninterpretable
What does it mean for all possible values from a Uniform Distribution to be “equiprobable”?
All the values have the same probability of occurring.
Values equally away from the mean are equally likely to occur.
All the possible outcomes are equal to the mean.
None of the above.
All the values have the same probability of occurring.
Why do the expected value and variance have no predictive power for a Discrete Uniform Distribution?
They do hold predictive power.
Because the expected value will posses the same probability as any of the other values in the sample space so we can’t create prediction intervals.
Because the standard deviation is not an integer value, so we cannot construct a prediction interval centered around the mean.
None of the above.
Because the expected value will posses the same probability as any of the other values in the sample space so we can’t create prediction intervals.
What are the two key characteristics of the Bernoulli Distribution?
A single trial and only two possible outcomes.
Several trials and only two possible outcomes.
Several trials and several possible outcomes.
A single trial and several possible outcomes.
A single trial and only two possible outcomes.
What is the relationship between Binomial and Bernoulli events?
A Binomial event is a sequence of identical Bernoulli events.
A Bernoulli event is a sequence of identical Binomial events.
There is no relationship between the two types of events.
None of the above.
A Binomial event is a sequence of identical Bernoulli events.
Why does the graph of the Poisson Distribution start from 0 on the X-axis?
Because no outcome can have a probability of occurring lower than 0.
Because we are measuring the frequency of occurrence over a given interval of time or distance.
The graph of the Poisson distribution does not start from 0 on the X-axis.
All of the above.
Because we are measuring the frequency of occurrence over a given interval of time or distance.
