Exam 1 Flashcards
statistics
scientific analysis of data in whose generation randomness/chance played some part
necessary to handle data since math isn’t enough because randomness is unavoidable in science
ex. how does the weight of a baby depend on its age? how does BP depend on genetics? Monty Hall Problem.
direction: probability is _____, while statistics is ______
prob is deductive. starts with if statement… then some prob calculation about data. ->
stat is inductive, starts with data and relevant prob calcs to make a statement about reality. <-
zig-zag relationship
probability notation
events: A, B, …
probability that event A will happen = Prob(A)
derived events
A u B: union, A or B or both
A n B: intersection, both
mutually exclusive events
F and G are mutually exclusive if they cannot happen in the same experiment. If F, G…H are mutually exclusive, P(F u G u … H) = Prob(F) + Prob(G)… + Prob(H).
independence
F and G are independent iff the probability of their intersection, Prob (F n G) = Prob(F) x Prob(G).
We often assume independence.
When F and G are independent, the conditional probability formula cancels out by filling in the prob of union equation. AKA F and G are independent if G having occured doesnt change the prob that F occurs.
If two events are mutually exclusive, they cannot be independent The oppositie may or may not be true.
conditional probabilities
the probability that F occurs given that G has occured
Prob (F | G) = Prob(F n G) / Prob(G)
discrete and continuous random variables
concept of the mind numbers
discrete RV: a conceptual and numerical quantity which in some future experiment will take on or other of a discrete set of values with known or unknown probabilities (only take one or other of a set of discrete values, usually 0, 1, 2, 3…)
continuous RV: can take any value in a continuous range of values RVs are written with upper case letters.
data
the observed values of RVs after the relevant experiment has been carried out. written with lower case letters.
presenting probabilities
tableau with possible values of X and probabilities of X (probs sometimes known, sometimes unknown)
or graph (known)
parameters
some usually unknown numerical value, denoted with a greek letter
discrete RV notation
Prob( X = vi) is a shorthand for “the prob that, once the exp is carried out, the observed value x of X will be vi”
conditions for binomial distribution
1) we conduct a fixed number of trials
2) there are only two possible outcomes on each trial. We call these success and failure
3) the outcomes of the various trials are independent
4) the probability of success, θ, is the same on all trials RV of interest is the number of successes, X, in n trials
third method of presenting probability distribtuion
(n x) is the # of orderings in which there are x successes in the n trials = n! / (n-x)!x!
the second part is the probability of getting x successes in n trials in a specified order
mean
NOT AN AVERAGE, a parameter, denoted μ, for binomal distribution, μ = nθ
it is the balance point (where areas under the curve are equal)
mean is a proberty of the being
MUST KNOW LONG AND SHORT FORMULAS FOR MEAN
average
calculated from data, estimates the mean, denoted x̄
precision depends on sample size and the propoerties of the RV X̄, the concept of the mind average before we roll the die
variance
always denoted σ2
it is a meausre of the spread outness of the probability distribution of X relative to the mean of X
for binomial distribution var of X = nθ (1 - θ)
important because the precision of x̄ as an estimate of μ depends on the variance of the RV X̄
MUST KNOW LONG AND SHORT FORMULAS FOR VARIANCE
complementary events
Ac is the event that A did not happen.
Prob (Ac) = 1 - Prob (A)