Discrete Distributions

Question 1

What is a distribution?

Answer 1

the possible values a variable can take and how frequently they occur

Question 2

What is the probability function?

Answer 2

p(y)

Question 3

What are probabilities or probability distributions?

Answer 3

measure the likelihood of an outcome

Question 4

We define distributions with which two characteristics

Answer 4

1. Mean & 2. Variance

Question 5

What character do we use to denote mean?

Answer 5

population: μ - mu

sample: x-bar

Question 6

What character do we use to denote variance?

Answer 6

population: σ2 - sigma squared

sample: s2 - s squared

Question 7

What character do we use to denote std deviation?

Answer 7

population: σ - sigma

sample - s

Question 8

What units are std deviation measured in?

Answer 8

The same units as the mean

Question 9

What units is variance measured in?

Answer 9

mean units squared

Question 10

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.

Answer 10

A collection of all the possible values a variable can take and how frequently they occur in the sample space.

Question 11

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.

Answer 11

Sample data represents some part of the data, while population data is the same as all the data.

Question 12

Which of the following is expressed with the letter s?

Population mean.

Population variance.

Population standard deviation.

Sample mean.

Sample variance.

Sample standard deviation.

Answer 12

Sample standard deviation.

Question 13

What are the Types of Probability Distributions?

Answer 13

1. Discrete Distributions - finite number of outcomes - examples: rolling a die or picking a playing card
2. Continuous distributions - infinitely many outcomes - example: Time and distance

Question 14

What is the notation for distributions?

Answer 14

X ~ N (μ, σ2)

Variable - Tilde - Type - Characteristics in parenthesis (usually mean, variance)

Question 15

What are some characteristics of Discrete Distributions?

Answer 15

• 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

Question 16

What is a Bernoulli Distribution?

Answer 16

• 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)

Question 17

What is a Binomial Distribution?

Answer 17

• The outcomes per iteration are two
• Many iterations
• a sequence of identical Bernoulli events
• Notation: B(n,p)

Question 18

What is the Poisson Distribution?

Answer 18

• 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

Question 19

What is Euler’s number? Napier’s Constant?

Answer 19

e ~ 2.72

Question 20

What are characteristics of Continuous Distributions?

Answer 20

The probability distribution graph would be a curve as opposed to unconnected individual bars

Question 21

What is the Normal Distribution?

Answer 21

outcomes often observed in nature

Question 22

What is the Chi-Squared? (Kai)

Answer 22

Asymmetric
only consists of non-negative values
always starts with zero on the left

Question 23

What is the Exponential distribution?

Answer 23

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

Question 24

What is the Logistic distribution?

Answer 24

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

Question 25

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.

Answer 25

Discrete and Continuous.

Question 26

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.

Answer 26

Because they share certain features.

Question 27

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.

Answer 27

We add up the probabilities of each individual outcome in the interval occurring.

Question 28

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)

Answer 28

P(Y<=y) = P(Y < y+1)

Question 29

What are some characteristics of a Uniform Distribution?

Answer 29

- 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

Question 30

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.

Answer 30

All the values have the same probability of occurring.

Question 31

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.

Answer 31

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.

Question 32

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.

Answer 32

A single trial and only two possible outcomes.

Question 33

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.

Answer 33

A Binomial event is a sequence of identical Bernoulli events.

Question 34

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.

Answer 34

Because we are measuring the frequency of occurrence over a given interval of time or distance.