Lecture 3 and 4

Binomial Distribution (assumptions)

- each trial only has 2 possible outcomes (dichotomous event) -the outcomes of trials are independent -the probability of success (p) is the same in every trial (across individuals)

Two outcomes are statistically independent if

P(A|B) = P(A | not B) = P(A) P(A and B) = P(A)P(B)

Bayes Rule

P(B|A) = P(A|B)P(B) / P(A); Allows you to flip the order of a conditional probability. Describes the probability of an event, based on prior knowledge of conditions that might be related to the event

Addition Rule (Not Mutually Exclusive)

Used to determine the probability that at least one of two events will occur. If the two events are not mutually exclusive; P(A or B) = P(A) + P(B) - P(A and B)

Binomial Distribution (mean)

mean = np

Poisson Approximation of Binomial

mu = np

multiplication Rule

"And"

Event

Specific way(s) the experiment can turn out.

Joint Probability

"And"; the probability of two events occurring together P(A and B)

Addition Rule

"Or"

Poisson distribution (assumptions)

1. The probability of an event in a short interval is proportional to the length of the interval 2. Rare (In any extremely small portion of the interval, the probability of more than one occurrence of the event is approximately zero) 3. Whether or not an event occurs in an interval is independent of events in all other intervals

Outcome

Exactly the experiment result.

Poisson Distribution (when to use)

For rare events. 3 uses: you are either given the expected number, rate or a time period to calculate the expected number, or its the approximation of the binomial distribution.

Addition rule (Mutually exclusive events)

If A and B are mutually exclusive events, P(A or B) = P(A) + P(B).

Probability

Provides a measure of the uncertainty associated with the occurrence of events. Probabilities describe "outcomes" of random experiments.

Theoretical Probability

The ratio of the number of equally likely outcomes in an event to the total number of possible outcomes. 𝑃(𝐸) = 𝑚/𝑁

Mutually exclusive events

Two events that cannot occur together (at the same time). The probability of their joint outcomes equals 0.

Poisson Distribution

a discrete probability distribution describing the likelihood of a particular number of independent events within a particular interval. Describes the totally random occurrence of events in time or objects in space. Often used to describe the distribution of the number of occurrences of a rare event.

Marginal Probability

the probability of a single event without consideration of any other event (independent of other events). Looking at totals of the columns or rows out of the overall total.

Conditional Probability

the probability of an event ( A ), given that another ( B ) has already occurred. Allows us to characterize and compare subpopulations. Key words: Among those, given. Looking at

Statistically independent events

the probability of one event occurring is unaffected by the occurrence or absence of the other. "Not associated." The probability of their joint outcome equals the product of the probabilities of occurrence of each outcome

Multiplication rule

the probability of two or more independent events occurring together can be determined by multiplying their individual probabilities. From the conditional probability, we can write the joint probability as P(A and B) = P(A|B) * P(B)