STATISTICS FOR BUSINESS AND ECONOMICS EXAM | QUESTIONS AND ANSWERS | VERIFIED ANSWERS (100% CORRECT) | LATEST EXAM UPDATE

STATISTICS FOR BUSINESS AND ECONOMICS EXAM | QUESTIONS
AND ANSWERS | VERIFIED ANSWERS (100% CORRECT) | LATEST
EXAM UPDATE




Experiment
An act or process of observation that leads to a single outcome that cannot
be predicted with certainty
sample points
the most basic outcomes of an experiment
Sample Space
The set of all possible outcomes.
two-way table
a statistical table that shows the observed number or frequency for two
variables, the rows indicating one category and the columns indicating the
other category. (responses are classified according to two variables)
Event
A subset of the sample space.
Simple Event
Events which consist of only one outcome.
Compound Event
Events which consist of multiple outcomes.
P = E₁ ∪ E₂ ∪ E₃ ...
Discrete Sample Space
A sample space which is either finite, or countably infinite.
- The relative probability of an event must be > 0
- Sum of the probability of all events = 1
A∪ B

,The set of outcomes in the event A, or B, or both.
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
A∩B
The set of outcomes in the event A and B.
P(A ∩ B) = P(A|B)P(B)
A^c
The set of outcomes that are not in A (A's complement).
P(A^c) = 1 - P(A)
∅
The empty event (an event that cannot happen).
P(∅) = 0
Mutually Exclusive
Two events that can't happen at the same time.
A∩B=∅
P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
P(S) - Probability of the Sample Space
P(S) = 1
If A ⊂ B...
P(B \ A) = P(B) - P(A)
P(A) ≤ P(B)
P(A) - Probability of an Event A
P(A) = P(A₁) + P(A₂) + P(A₃)..., where A₋n is an elementary subset of the
event A.
In an equi-probably sample space, P(A) = |A| / |S|.
|A|
The number of outcomes in an event A.
List the Sample Space Method
Find the probability of an event by listing the sample space, then dividing
the number of elements in the event by the total number of elements in the
samples space.
Basic Principle of Counting

,If operation 1 can be done in m ways, and operation 2 can be done in n
ways, then the combined operation can be done in m * n ways.
Permutations
The number of ordered arrangements of n distinct objects, taken r at a
time.
nPr = n! / (n - r)!
Combinations Rules
A sample of r unordered elements is to be drawn from a set of n elements,
the number of different samples possible is denoted by:
nCr or (n/r)= n! / r!(n - r)! ! is factorial symbol
Partitioning Objects into Distinct Groups (where each group is of a given
size)
To partition n objects into k distinct groups, with each group containing n₋i
objects:
n! / n₁!n₂!n₃!...n₋k!
P(A|B) - Probability of Event B given that an Event A has occurred
P(A|B) = P(A ∩ B) / P(B)
P(B|B)
P(B|B) = 1
P(A^c|B)
P(A^c|B) = 1 - P(A|B)
P(C ∪ D|B)
P(C ∪ D|B) = P(C|B) + P(D|B)
Independent Events
Two events are independent if P(A ∩ B) = P(A)P(B).
In independent events:
P(B|A) = P(B)
P(A|B) = P(A)
Three events are independent if any two of them are independent: P(A ∩ B
∩ C) = P(A)P(B)P(C)
Partitions

, Multiple events form a partition of the sample space if they are pairwise
mutually exclusive, and the union of the events is the entire sample space.
Law of Total Probability
P(A) = ∑(i=1 to n) P(A|B₋i)P(B₋i)
Bayes Rule
The probability of a subset B₋k of a partition, given an event A:
P(B₋k|A)
= P(B₋k ∩ A) / P(A)
= P(A|B₋k)P(B₋k) / ∑(i=1 to n) P(A|B₋i)P(B₋i)
Random Variable
If X is a random variable, then P(X = x) is the probability of seeing a
specific value in the sample space.
Discrete Random Variable
A random variable where the range of values that the variable can take on
is a countable set
Probability Function
f(x) = P[X = x], x ∈ Rx (the range of values x can take on)
- f(x) ≥ 0 for all x
- ∑ f(x) = 1
Expected Value of a DRV
E(x)
= ∑(x ∈ Rx) x * f(x)
= ∑(x ∈ Rx) x * P[X = x]
E[g(x)] = ∑(x ∈ Rx) g(x) * f(x)
E[g₁(x) + g₂(x)] = E[g₁(x)] + E[g₂(x)]
E[cg(x)] = cE[g(x)]
Variance of a DRV
Var(x)
= E[(X - µ)²]
= ∑(x ∈ Rx) (x - µ)² * f(x)
= E(X²) - µ²
Bernoulli Random Variable