2: Statistical Review

Question 1

sample space

Answer 1

set of all possible outcomes

Question 2

event

Answer 2

subset of the sample space

Question 3

random variable

Answer 3

variable whose possible values are numerical outcomes of some random phenomenon

function defined on the set of possible outcomes that assigns a real number to every possible outcome

Question 4

two major classes of random variables

Answer 4

discrete and continuous

Question 5

probability distribution

Answer 5

number between 0 and 1 that quantified how likely an event is to occur

Question 6

probability function

Answer 6

describes/characterises discrete random variable

probability for each possible discrete outcome

Question 7

cumulative distribution function

Answer 7

describes/characterises distribution of a random variable

lists the probability that a random variable is less than or equal to a specific value

also called the distribution function or cumulative risk profile

Question 8

continuous random variable

Answer 8

random variable that can take on any real value within some range

Question 9

probability density function

Answer 9

determines probabilities associated with continuous random variable

Question 10

mode

Answer 10

value occurring with the greatest probability

Question 11

median

Answer 11

value such that the probability of the random variable being less than or equal to that value is at least 50% and the probability of the random variable being greater than or equal to that value is at least 50%

Question 12

mean/expected value

Answer 12

weighted average of all possible outcomes, weighted by probabilities of outcomes

Question 13

variance

Answer 13

measures the spread or dispersion of the variable around its mean

standard deviation^2

Question 14

characteristics of normal distribution

Answer 14

defined by mean and standard deviation

single-peaked

symmetric around the mean

standardised to have mean 0 and variance 1

Question 15

joint probability distribution

Answer 15

probability that two random variables can simultaneously take on particular values

Question 16

conditional distribution

Answer 16

distribution of a random variable conditional on another random variable taking on a specific value

Question 17

conditional expectation

Answer 17

expected value of a random variable, conditional on the realised value of another random variable

Question 18

independence

Answer 18

X ⊥Y

knowing X tells you nothing about Y

going distribution is the product of the marginals

Question 19

covariance

Answer 19

variance of X - measures how X alone varies

covariance of X and Y - measures how X and Y vary together

Question 20

correlation

Answer 20

covariance rescaled between -1 and 1 (unit-free)

Question 21

independence vs uncorrelated

Answer 21

independent random variables are also uncorrelated, but the reverse is not true

Question 22

population

Answer 22

set of all information of interest to the decision-maker

Question 23

sample

Answer 23

subset of a population

to be useful, has to be representative of the population

Question 24

single random sample

Answer 24

if each individual population is equally likely to be included in the sample

e.g. random draws

leads to independent and identically distributed draws

Question 25

point estimates

Answer 25

estimator computed from sub-sample for the sample of data which is a subset of the population to learn about the population

single number used to estimate an unknown population parameter

Question 26

the law of large numbers

Answer 26

as n goes to infinity, the expected value of the mean will be very close to the population mean

Question 27

the central limit theorem

Answer 27

the average from a random sample for any population (with finite variance), when standardised, has an asymptotic standard normal distribution, meaning that it becomes well approximated by a standard normal

Question 28

properties of estimators

Answer 28

unbiasedness, consistency, efficiency