STATS midterm 2

implied probability

expectation vs function

variance and sd are not

linear

bernoulli pmf

bernoulli expectation + var

bernoulli explanation

Simplest RV with only 2 outcomes, control parameter p to see probability of success

building block of discrete probability, modeled as X ~ Bernoulli(p)

E[X] = p will tell you probability of success, p also says expected value

variance p(1 - p) captures max uncertainty in middle of distribution

joint pmf properties

you cannot calculate joint pmf of (X , Y ) from the marginals

but you can calculate the marginals from the joint!

strong linearity

if x and y are independent correlation is

0

expectation of sum = sum of expected values

expectation v variance rule

conditional independence

changes for continuous RVs

continuous probability

uniform 0,1 RV

properties of CDF

probability density function

pdf and cdf

pdf properties

probability of intervals in continuous RV

variance continuous RV

continuous independence

expectation rules for multiple RVs

standard normal distribution X~N(0,1)

N(0, 1) PDF and CDF notation

dnorm for N(0,1)

pnorm N(0,1)

rnorm N(0,1)

expectation and variance N(0,1)

general normal dist N(0, 1)

mew as expectation

location shift normal pdf

sigma variance

sigma dispersion

N mew sigma pdf

n mew sigma cdf

n mew sigma draws

sum of multiple normal RVs is normal

linear function of normal RV

answer is B

interval probability of N mew sigma

quantiles of continuous RVs

Threshold right tail probability N(0,1)

symmetric intevals N01 RV

check

3 sigma rule

z score 3 sigma

iid

independent and individually distributed

mutually independent and same identity distribution

sample stats

combine summary stats and probability theory

random sampling

“equally-likely sampling with replacement”, i.e.

(i) each unit of the population is equally likely to be selected;

(ii) each selection is independent from other selections.

random sampling as iid rvs

var x bar n

sampling dist x bar n

sample dist x bar n w/o normality

law of large numbers

formal LLN

central limit theorem

formal CLT

estimator

description of a general procedure

estimate

output of procedure

sample mean x bar n is estimator

of expectation mew

parameter and estimator

sample variance uses

1/(n - 1)

variance estimator

The variance of bθ is just the variance of ˆθ as a RV.

▶ Var bθ is defined without reference to the true parameter θ0.

↔ compare with the definition of bias, which involves θ0.

estimator efficiency

tradeoff between bias and variance:

− low bias estimators often have a high variance

− low variance estimators often have high bias

mean squared error of estimator

expected squared distance between the estimator ˆθ and the

true value θ0.

smaller = better

finite sample properties

for each particular sample size n, what are the properties of the

sampling distribution of bθn?

− e.g. bias, variance, MSE, finite-sample sampling distribution...

asymptotic large sample properties

what happens to the sampling distribution of bθn as the sample

size n gets larger and larger?

− what are the limits of (the properties of) the sampling

distribution of bθn as n → ∞?

− e.g. various limits of bias, variance, MSE, distribution...

consistency

LLN Consistency

consistency of plug in estimators

recycle plug in estimators

why sample var is consistent

asymptotic dist. of an estimator

central limit theorem

CLT and asymptotic distribution

p and arrow

convergence in probability

is about the convergence of RVs to a constant

→ there is no randomness in the limit

→ e.g. LLN

d and arrow

convergence in distribution

is about convergence of the distributions (of RVs) to another

distribution.

→ there is randomness in the limit:

→ e.g. CLT

confidence interval

confidence interval is a sample statistic

want to be more confident?

what affects ME and CI

CI interpretation

more on CI

probability is about sampling dist. of CI

if we sampled many times we’d get many

different sample means, each leading to a different confidence

interval. Approximately 95% of these intervals will contain μ.

confidence interval summary

important confidence interval summary

normal meaning

A Normal (or Gaussian) distribution is a specific shape that a random variable's distribution can take. It shows up constantly in statistics because of how naturally it arises in the real world.