Formula for when you want to have 'r' objects from 'n' with replacement, the order is not important
ex: Consider the lottery, pick three numbers from 1,2,3,4 with replacement
What is conditional probability?
updating one's beliefs based on obtaining new knowledge
Formula for finding the probability of A, given that B has ocured:
P(A|B)
If A and B are disjoint, A ∩ B = ∅,
then P(A|B)=?
0
Multiplication Law
Let A and B be events with P(B)≠0
if P(A)>0, likewise we have:
Law of Total Probability
Given a partition A1, A2, ... with P(Ai) > 0 for all i
Bayes' Rule
Independent Events-
Two events A and B, each with positive probability, are independent if any of the following are true:
Describe what these equalities imply, if true:
Independence
For two events A and B, if any of those equalties are true, then the two events are independent
hypothesis test idea in terms of independence:
Assume independence (null hypothesis) between the variables and see if the data refutes the assumption
Χ2 test:
How can we quantify differences between observed and expected cell counts?
With a chi-squared test statistic!
Χ2
More independence truths, about events and their compliments:
With two events A and B:
What is an alternate way to write P(A)?
T/F Two events can be both disjoint and independent
False
Define Inference:
making claims about our population (parameters) using sample data with mathematical rigor
Practically speaking, what does the law of large numbers tell us?
This tells us that the sample mean (or mean of any RVs) that have the same mean and variance will be observed closer and closer to the ’population’ mean as the number of things you are averaging increases.
Define permutation:
an ordered arrangement of objects
Define: multiplication principle
If one experiment has m outcomes and another experiment has n outcomes, then there are mn possible outcomes for the two experiments
How are permutations different from combinations?
For permutations, we are no longer interested in ordered samples, but in the constituents of the samples regardless of the order in which they were obtained.
