week 1

Question 1

Probability

Answer 1

: a branch of mathematics concerning the analysis of random phenomena.

Question 2

Random phenomena

Answer 2

processes with an uncertain outcome.
(e.g., flipping a coin; gambling games)

Question 3

Inferential statistics and probability are related because…

Answer 3

sampling a group of people from the population is a random phenomenon.

Question 4

In probability, we know the true model/mechanism in the population. Based on the true model, we compute…

Answer 4

the probability of different outcomes.

e.g., If I flip a fair coin 10 times, how likely is it that I will get 5 heads?

Question 5

In inferential statistics, we do NOT know…

Answer 5

the true model/mechanism in the population. We infer the true model-based on the outcomes from our sample data

e.g., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? In other words, is the coin a fair coin?

Question 6

In probability, the term experiment is used in a loose sense to mean…

Answer 6

a procedure for which the outcome is uncertain.

Examples of experiments include:
§ an experimental study
§ toss of a coin

Question 7

sample space of the experiment

Answer 7

The set of all possible outcomes of an experiment

is denoted by S.

§ Best to think of the sample space as an area.

Question 8

random event or an event

Answer 8

A subset of the sample space

If the experiment consists of flipping two coins, then an event can be getting head on the first coin:

Question 9

Probability measure

Answer 9

function that maps the random events in
the sample space onto the real numbers between 0 to 1.

The function “measures” the area of the event out of the whole sample space.

Question 10

Probability of an event E is denoted as

Answer 10

P(E)

Question 11

Frequentist and Bayesian perspectives have different conceptualizations of…

Answer 11

the probability measure.

different views on how we should map the events in the sample space onto the real numbers between 0 and 1.

Question 12

N(E) represents the…

Answer 12

number of times in the first N repetitions of the experiment that the event E occurs.

Question 13

In the frequentist perspective, what is the probability of an event?

Answer 13

The probability of the event is the proportion of times the event E has occurred as we perform the same experiment infinitely many times (i.e., N reaches infinity).

probability is the frequency of the event
occurrence, hence called the frequentist perspective.

Question 14

In the Bayesian perspective, what is the probability of an event?

Answer 14

represents a degree of your subjective belief about the occurrence of an event

Question 15

frequentist definition

Answer 15

long-run probability

Question 16

bayesian

Answer 16

degree of belief

Question 17

Properties of frequentist perspective

Answer 17

Objective/Unambiguous

Can’t assign probability to events that are not replicable

Question 18

Properties of bayesian perspective

Answer 18

subjective/ ambiguous

can assign probability to any event

Question 19

What is a random variance?

Answer 19

A random variable is a function that maps random events in the sample space of an experiment onto the real number line.

Through a random variable, we can use numbers to quantify
or represent the occurrence of an event.
-usually denoted by a capital letter (e.g., X or Y )
- different from the algebraic variable (e.g., a ` 5), which means any unspecified number.

Question 20

An indicator (or Bernoulli) random variable (X) maps…

Answer 20

the occurrence of the event to 1.
the non-occurrence of the event to 0

Question 21

How to denote a bernoulli random variable:

For example, let X indicate whether we get a head after a coin flip.

Answer 21

X(H) = 1
X(T) =0

Question 22

Discrete random variables

Answer 22

can only take on specific values, usually whole numbers

indicator random variable (X “ 0, 1); binomial random variable
(X “ 0, 1, 2, 3 . . .)

countable number of values.

Question 23

Continuous random variables

Answer 23

can take on any value in an
interval

e.g., normal random variable.Can take on any value on the real number line from positive to negative infinity
X = 0.00001

uncountable number of values.

Question 24

What does the probability measure of the random variable map?

Answer 24

For a random variable, the probability measure maps the values of the random variable onto a value between 0 and 1, which measures the likelihood of the values of the random variable.

Question 25

probability distribution.

Answer 25

Each random variable has a probability distribution. Discrete: probability mass function (PMF) § Tells us the probability associated with each possible value of the random variable. § Continuous: probability density function (PDF)

Question 26

Probability mass function of random variable

Answer 26

In the example of X being the indicator random variable representing getting a head after a fair coin flip, the PMF of X is P(X=0) = 0.5 P(X=1) = 0.5

Question 27

Bernoulli Distribution

Answer 27

If a random variable is a Bernoulli random variable, we can say that the random variable follows the Bernoulli distribution. By Bernoulli distribution, we mean the probability distribution associated with the Bernoulli random variable.

Question 28

If X is a Bernoulli random variable where P(X=1) = p then we can write

Answer 28

X ~Ber(p) where the symbol “~” stands for “follows”, and Ber stands for Bernoulli distribution.

Question 29

For brand-named random variables, their distributions are characterized by a small number of parameters. Explain parameter in this context

Answer 29

For X ~Ber(p), p is the parameter that fully describes the Bernoulli distribution. Parameters are considered non-random, fixed variables. This usage of the term “parameter” is a bit different but related to the case when “parameter” is used to mean the quantities computed with population data.

Question 30

Normal distribution

Answer 30

A normal random variable (a.k.a., Gaussian random variable) is a continuous random variable that follows the famous “bell curve” distribution.

Question 31

The “bell curve” distribution is called the __________________________________________________________ of the normal random variable

Answer 31

The “bell curve” distribution is called the Probability Density Distribution (PDF) of the normal random variable

Question 32

The normal random variable is characterized by two parameters:

Answer 32

1. expected value u 2. variance o^2

Question 33

How to denote X as a normal random variable:

Answer 33

X ~ N (u,o^2)

Question 34

Standard Normal Random Variable and how to denote it

Answer 34

When the normal random variable has a mean of 0 and a variance of 1, then it is called the standard normal random variable, usually denoted as Z ~ N (0,1)

Question 35

We can transform any normal random variable to the standard normal variable. Then you can transform X to follow the standard normal distribution by

Answer 35

Z = X-u/o

Question 36

what is the probability of a Continuous Random Variable taking on any specific value ?

Answer 36

For a continuous random variable, we cannot talk about the probability of the random variable taking on any specific value. the probability of a continuous random variable taking on a specific value is always zero. For a continuous random variable, we can only talk about the probability of the random variable taking on a range of possible values.

Question 37

cumulative distribution function (CDF)

Answer 37

tells us the probability of a random variable taking on a value that is equal to or less than a cutoff point. P(X< a) or P(X < a) is the area under the curve below a

Question 38

68–95–99.7 Rule

Answer 38

The 68–95–99.7 rule is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution.

Question 39

There are four R functions for the normal distribution:

Answer 39

dnorm() pnorm() qnorm() rnorm()

Question 40

dnorm()

Answer 40

The dnorm() function computes the PDF of the normal distribution. Output the probability density of a normal random variable at a specific value Not commonly used because for continuous random variables, the probability of a range of values is more important (i.e., the area under the PDF)

Question 41

pnorm()

Answer 41

The pnorm() function computes the CDF of the normal distribution Output the probability of a normal random variable taking on values below the quantile value. Need to input: q: the quantile value at which you want to compute the probability. mean: value for the parameter µ. sd: value for the parameter σ. Other input: § lower.tail: logical; whether you want the upper tail or the lower tail probability. By default, lower.tail=T.

Question 42

qnorm()

Answer 42

The qnorm() function computes the quantile value given a probability below the quantile value. Output the quantile value. Need to input: p: the probability below the quantile value. mean: value for the parameter µ. sd: value for the parameter σ. Other input: lower.tail: logical; whether you specified the upper tail or the lower tail probability for p. By default, lower.tail=T.

Question 43

In the R functions what do you need to remember about the variance?

Answer 43

Note: Remember to square root the variance to get the standard deviation for the argument sd.

Question 44

rnorm()

Answer 44

generates/simulates random numbers from the normal distribution Suppose our population data follow a normal distribution N(100, 400). We want to simulate randomly sampling 10 values from the population. Then we can do rnorm(n = 10, mean = 100, sd = sqrt(400))

Question 45

Binomial distribution: Notation? What kind of random variable?

Answer 45

X ~ Bin(N,p) discrete random variable

Question 46

Chi-square distribution: Notation? What kind of random variable?

Answer 46

X ~ x^2(df) continuous random variable

Question 47

t distribution: Notation? What kind of random variable?

Answer 47

X ~ t(df) continuous random variable

Question 48

Random variables characteristics

Answer 48

Associated with random events. Have probability distribution Can take on more than one possible value. Denote using capital letters XY

Question 49

Constants or Fixed Values

Answer 49

Associated with non-random event Do not have probability distribution Can only take on one possible value Denote using small letters ax

Question 50

What does a random variable quantify?

Answer 50

a random procedure’s different outcomes.

Question 51

once you see the random procedure’s outcome, it is called....

Answer 51

the realized value of a random variable. The realized value of a random variable is treated as constant

Question 52

empirical probability distribution.

Answer 52

We can also realize this random variable multiple times and then graph the empirical probability distribution We can realize the random variable 10 times by flipping a fair coin 10 times. The empirical probability distribution is an estimation of the theoretical probability distribution.

Question 53

Usually, the population data of a variable are assumed to follow....

Answer 53

the normal distribution

Question 54

Sample statistics (e.g., the sample mean) across repeated studies are _____________________ __________

Answer 54

random variables - has a probability distribution

Question 55

Population parameters (e.g., the population mean) are __________________

Answer 55

constants do not have a probability distribution

Question 56

The sample data are random across repeated sampling; therefore, sample statistics are also ___________

Answer 56

random

Question 57

Population parameters are considered _____________________________ in the Frequentist perspective.

Answer 57

constants (or fixed values)

Question 58

Do population parameters have any probability distributions associated?

Answer 58

No bc they are constants

Question 59

Parameters of a random variable:

Answer 59

numerical quantities that fully describe a distribution u and o^2 in X ~ N (u,o^2)

Question 60

Population parameters:

Answer 60

numerical quantities characterizing the population data

Question 61

From the Bayesian perspective, population parameters are considered...

Answer 61

random variables because we are uncertain about their values. In Bayesian statistics, you can specify a probability distribution for each parameter. § called prior distribution.

Question 62

CLT roughly implies what?

Answer 62

that when we add or average a large number of random variables, the sum or the mean of the random variables is a random variable that follows a normal distribution. CLT implies when you add or average different random events together and use a random variable to quantify it, then the probability measure of the random variable follows the normal distribution

Question 63

CLT formula

Answer 63

At a large n, Xbar approximately follows a normal distribution N(uxbar = u, o2/x = o^2/n)

Question 64

uxbar

Answer 64

the mean of the sampling distribution of sample mean xbar

Question 65

o

Question 66

oxbar

Answer 66

the standard deviation of the sample distribution of the sample mean X; standard error of the mean SEM

Question 67

In essence, the CLT roughly implies

Answer 67

that when we add oraverage a large number of random variables each with finite µ and σ2 the sum or the mean of the random variables follows a normal distribution. This implies when you add different random events together and map them onto a number line, it follows the normal distribution.

Question 68

One of the most common applications of the CLT is regarding

Answer 68

the sampling distribution of the sample mean.

Question 69

What is the sampling distribution of the sample mean

Answer 69

The sampling distribution of the sample mean is the distribution of the sample mean over repeated samples. § “Over repeated samples” means “conducting the same experiment (with a fixed sample size n) infinitely many times.” § Related to the frequentist perspective.

Question 70

according to CLT, the sampling distribution of the sample mean is a ______________ distribution

Answer 70

normal distribution.