STAT Exam 2

expected value

weighted average, where the probability distribution of a discrete random variable serves as the weights in the weighted average.

complement

The complement of A is the set of all outcomes in the sample space that don't belong to A

independence

The trials are mutually independent. The result of any one trial has no effect on the results of any of the other trials.

Fixed Number of Trials

There are a total of n trials (or repetitions) of an experiment. Each trial ends in one of two complementary events, A or Ac, where event A corresponds to success and event Ac corresponds to failure.

False-Positive Rate

1 − P(NEG|Dc) = P(POS|Dc).

How is a probability histogram constructed?

1. For a given value x, a rectangle with base [x−0.5, x+0.5] and height P (X = x) is drawn. Since the width of the base is 1, the area equals P (X = x). 2. The process is repeated for all values of X. The sum of the areas must be 1.

Pairwise disjoint event notation

Ai ⋂ Aj = 0, whenever I ≠ j

Not

Complement (c)

Events A and B are said to be negatively associated if

P(A ∩ B) < P(A)P(B)

Range rule

The probability of any event is a number between 0 and 1. 0 <​ P (A) <​ 1 when A ⊆ S.

OR

Union (U)

For

conditional

discrete

if its values form a finite set of numbers or a sequence of numbers such as 0, 1, 2, ....

Mean

Greek letter μ ("mu")

standard deviation of X.

Greek letter σ ("sigma")

P (X = x)

denote the probability that outcome x occurs.

AND

intersect

False-Negative Rate

1 − P(POS|D) = P(NEG|D).

Disjoint Event notation

A⋂B = 0, where "0" is the empty set

Intersection Rule

If A and B are events and P(B) ≠ 0, then the probability of the intersection of the events is the probability of B times the probability of A given B

Intersection Rule for Independent Events

If A and B are independent, then the probability of the intersection of the events is the probability of A times the probability of B: P(A ∩ B) = P(A)P(B) when A and B are independent events

Subset Rule

If A is a subset of B, then the probability of A must less than or equal to the probability of P(A) ≤ P(B) when A ⊆ B ⊆ S.

The probability of the intersection of the two events is equal to the product of the probabilities of the events A, B,

P(A ∩ B) = P(A)P(B)

Formula for intersection probability.

P(A ∩ B) = P(B)P(A|B).

Events A and B are said to be positively associated if

P(A ∩ B) > P(A)P(B)

The conditional probability of A given B is the same as the probability of A

P(A|B) = P(A)

formula for conditional probability

P(A∩B) P(A|B)= P(B) .

Rule of Average Conditional Probabilities:

P(B) = P(A1)P(B|A1)+P(A2)P(B|A2)+···+P(Ak)P(B|Ak), when P(Ai) ̸= 0 for each i.

The conditional probability of B given A is the same as the probability of B

P(B|A) = P(B)

pairwise disjoint events

Pair A1, A2...Ak are said to be pairwise disjoint (or pairwise mutually exclusive) if each pair of events is disjoint

Partition notation

S = A1 ⋃ A2 ⋃....Ak where Ai ⋂ Aj = 0 when i ≠ j

sample space

S is the set of all possible outcomes of an experiment

A ⋂ B

Set of outcomes that belong to both A and B

Complement Rule

The probability of the complement of an event is 1 minus the probability of the event P(Ac) = 1 − P(A) when A ⊆ S.

Disjoint Union Rule

The probability of the union of mutually exclusive events is the sum of the probabilities of each event: P (A ∪ B) = P (A) + P (B) when A ∩ B = ∅.

Union Rule

The probability of the union of two events that are not mutually exclusive is the sum of the probabilities of each event minus the probability of their intersection: P (A ∪ B) = P (A) + P (B) − P (A ∩ B) when A ∩ B ̸= ∅.

Constant Probabilities:

The success probability, p = P(A), is constant from trial to trial.

factorial

a descending product of integers defined for nonnegative integers. Ifn=0, then n!=0!=1.

probability histogram

a graphical display of the probability distribution of a discrete random variable with whole number values, where area is used to represent probabilit

(A⋂B)C = Ac⋃Bc

complement of the intersection equals the union of the complements

(A⋃B)c = Ac ⋂ Bc

complement of the union equals the intersection of the complements

Of

conditional

Probability of event A

denoted by P(A) ("probability of A")

sensitivity

diagnostic test is the probability that an individual chosen at random from those with disease tests positive for disease Sensitivity = P(POS|D).

Conditional distributions

distribution of one variable given a fixed value of another variable.

Disjoint Events

events A and B are said to be disjoint (or mutually exclusive) if they have no outcomes in common

conditional expected value

expected value computed using a conditional probability distribution.

Hypergeometric Distribution

gives the probability distribution of the number of individuals from a subpopulation of interest in a simple random sample from a study population.

binomial distribution

gives the probability distribution of the number of successes in a fixed number of trials of an experiment

But

intersection

partition

is a collection of pairwise disjoint (or pairwise mutually exclusive events A1, A2,....Ak whose union is S.

probability model or probability distribution

is a specification of numbers P(A) that satisifies probability rules

event

is a subset of the sample space s

x

number of individuals from the subpopulation of interest in a given simple random sample of size n, and (n-x) is the number of individuals from the complementary subpopulation in the sample/

n

number of individuals in the simple random sample

joint distribution

probability distribution of two or more random variables.

conditional probability

probability of an event occurring given that certain conditions hold. P(A|B) (and say "the probability of A given B").

0-1 Rule

probability of the empty set is 0 and the probability of the full sample space is 1: P(0) = 0 and P(S) = 1

specificity

probability that an individual chosen at random from those who are disease free tests negative for disease Specificity = P(NEG|Dc).

Prevalence

probability that an individual chosen at random from the study population has the disease P(D)

P(A)

proportion of times event A will occur in a sufficiently long series of repetitions of the experiment

A⋂ Bc

set of outcomes that belong to A but not to B

Ac ⋂ B

set of outcomes that belong to B but not to A

Ac ⋂ Bc

set of outcomes that belong to neither nor B

probability distribution of X

specifies all possible outcomes of the random variable and gives the probability that each will occur

intersection

the intersection of events A and B is the set of outcomes that belong to both A and B. The notation A∩B is used to denote the intersection

Predictive Value of a Negative Test

the probability that a person with a negative test does not have the disease. P(Dc|NEG)

predictive value of a positive test

the probability that a person with a positive test is a true positive (i.e. does have the disease). P(Dc|NEG)

independence

the trials are mutually independent. The result of any one trial has no effect on the results of nay other trials

Union

the union of events A and B is the set of outcomes that belong to either A or B. The notation A U B is used to denote the union

N

total number of individuals in the study population

M

total number of individuals in the study population of interest, and N - M is the total number of individuals in the complementary subpopulation

experiment

used in probability theory to describe a procedure whose outcome is not known in advance with certainty

random variable

variable whose value is a numerical outcome of a random phenomenon.