Revision

Question 1

Formula for probability with replacement where all outcomes equally likely

Answer 1

P(A) = n(A)/n(S)

where A is the event

n(A) is the number of different ways the event can happen

n(S) is the number of different ways things can happen in sample space

Question 2

Formula for probability without replacement where all outcomes equally likely

Answer 2

Same as for with replacement: P(A) = n(A)/n(S)

except use factorials not powers now to account for not replacing

Question 3

permutation

Answer 3

Any ordered arrangement of objects

Question 4

The number of different permutations of N objects, e.g. number of permuations of three animals

Answer 4

The number of different permutations of N objects is N!

So number of three animals is 3! = 6

fox, rabbit, dog

fox, dog, rabbit

rabbit, fox, dog

rabbit, dog, fox

dog, rabbit, fox

dog, fox, rabbit

Question 5

The number of different permutations of n objects taken from N, e.g. 2 animals from 3 animals

Answer 5

The number of different permutations of n objects taken from N objects is N!/(N-n)!

e.g. 2 animals from 3 animals = 3!/1! = 6

fox,rabbit

rabbit,fox

fox,dog

dog,fox

rabbit,dog

dog,rabbit

Question 6

Combination

Answer 6

Any unordered arrangement of objects is called a combination

Question 7

The number of different combinations of n objects taken from N objects,

e.g. 2 animals from 3

Answer 7

The number of different combinations of n objects taken from N objects is N!/{(N-n)!n!} written as N choose n (N n)^T

e.g. 2 animals from 3 = (3 2)^T = 3!/(3-2)!2! = 3!/2! = 3

dog, rabbit

fox,rabbit

dog,fox

Question 8

hypergeometric experiment

Answer 8

A hypergeometric experiment is a statistical experiment that has the following properties:

• A sample of size n is randomly selected without replacement from a population of N items.
• In the population, k items can be classified as successes, and N - k items can be classified as failures.

Question 9

hypergeometric probability

Answer 9

h(x; N, n, k): hypergeometric probability - the probability that an n-trial hypergeometric experiment results in exactly x successes, when the population consists of N items, k of which are classified as successes.

Question 10

binomial experiment

Answer 10

A binomial experiment is a statistical experiment that has the following properties:

• The experiment consists of n repeated trials.
• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
• The probability of success, denoted by P, is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Question 11

binomial probability

Answer 11

The binomial probability refers to the probability that a binomial experiment results in exactly x successes.

Question 12

Key principles of the Belmont report

Answer 12

1 Respect for persons:

• Respect individual autonomy
• Protect individuals with reduced autonomy

2 Beneficence:

• Maximize benefits and minimize harms

3 Justice

• Equitable distribution of research burdens and benefits

Question 13

Conditional probability formula

Answer 13

Question 14

Bayes Rule

Answer 14

Question 15

mean and variance of binomial

Answer 15

mean of binomial np

variance of binomial: nP*(1-P).

Question 16

uniform distribution

Answer 16

uniform distribution refers to a probability distribution for which all of the values that a random variable can take on occur with equal probability

P(X=x_k) = 1/k where there are k different values, e.g. 1/6 for probability of rolling a 4

Question 17

mean of hypogeometric distribution

Answer 17

nK/N

Question 18

Variance of hypogeometric distribution

Answer 18

Question 19

Difference between machine learning and RCTs

Answer 19

Machine learning conerned with prediction;

RCT concerned with causal estimation

Question 20

Mean of exponential function

Answer 20

1/λ

Question 21

mean of uniform distribution, e.g. U~(1,15)

Answer 21

Mean of uniform distribution: (a+b)/2

e.g. mean of U(1,15)=(1+15)/2=8

Question 22

95% confidence interval formula for sample mean where s² is sample variance and n=10

Answer 22

Question 23

Variance in a standard Neyman analysis

Answer 23

Question 24

Fisher exact test

Answer 24

The Fisher exact test tests the hypothesis that the treatment effect is identically 0 for all treatment units.

Question 25

90 confidence interval z score

Answer 25

1.645

Question 26

95% confidence interval z score

Answer 26

1.96

Question 27

99% confidence interval z score

Answer 27

2.576

Question 28

CI from table of results

Answer 28

point estimate +- z\*s.e.(point estimate)

Question 29

2sls first-stage equation and second-stage equation Y= a +bX + e

Answer 29

Regress X on Z then plug Xhat into original equaiton

Question 30

Assumptions are needed for an instrument to be a good instrument Y = a + bX + e

Answer 30

endogenous variable X highly correlated to instrument Z Z not correlated to e Z does not affect Y accept through X

Question 31

Poisson experiment

Answer 31

A Poisson experiment is a statistical experiment that has the following properties: * The experiment results in outcomes that can be classified as successes or failures. * The average number of successes (μ) that occurs in a specified region is known. * The probability that a success will occur is proportional to the size of the region. * The probability that a success will occur in an extremely small region is virtually zero.

Question 32

Poisson probability

Answer 32

P(x; μ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ. P(x; μ) = (e^-μ) (μ^x) / x!

Question 33

Mean and variance of Poison distribution

Answer 33

The mean of the distribution is equal to μ . The variance is also equal to μ .