Probability
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The field of probability, within mathematics, assesses the chances of an event occurring.
This likelihood is represented by a numerical value ranging between 0 and 1, where higher values indicate a greater probability.
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Event
Events are sets of outcomes in experiments with assigned probabilities. A single outcome can belong to many different events,
which are not equally likely since they can include different groups of outcomes.
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Likelihood
A measure of how well a statistical model explains the observed data. In probability,
it often refers to the probability of an interval rather than an individual value.
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Outcome
A single result from a probability experiment. Each distinct outcome has a probability assigned to it based on certain criteria,
such as the size of its associated interval in a probability distribution.
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Preferred Outcomes
Preferred outcomes are the outcomes we want to occur or the outcomes we are interested in. We also call refer to such
outcomes as “Favorable”.
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Sample Space
Sample space refers to all possible outcomes that can occur. Its “size” indicates the amount of elements in it.
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Probability Formula
The Probability of event X occurring equals the number of preferred outcomes over the number of outcomes in the
sample space.
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Trial
Observing an event occur and recording the outcome.
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Experiment
A collection of one or multiple trials.
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Experimental Probability
The probability we assign an event, based on an experiment we conduct.
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Expected Value
The weighted average of all possible values that a random variable can take on.
It is the specific outcome we expect to occur when we run an experiment.
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Expected value for categorical variables.
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Expected Value (Numerical Variables)
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Frequency
Frequency is the number of times a given value or outcome appears in the sample space.
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Frequency Distribution Table
The frequency distribution table is a table matching each distinct outcome in the sample space to its associated frequency.
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Probability Frequency Distribution
A collection of the probabilities for each possible outcome of an event.
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Complement
The complement of an event is everything an event is not.
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Characteristics of Complements
• Can never occur simultaneously.
• Add up to the sample space. (A + A’ = Sample space)
• Their probabilities add up to 1. (P(A) + P(A’) = 1)
• The complement of a complement is the original event. ((A’)’ = A)
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The probability of an event A is 0.25.
What is the probability of A’ (the compliment of A)?
P(A’) = 0.75
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Combinatorics
A branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints.
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Factorials
Factorials express the product of all integers from 1 to n and we denote them with the “!” symbol.
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Key Values for Factorials
• 0! = 1.
• If n<0, n! does not exist.
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Rules for Factorial Multiplication. (For n>0 and n>k)
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Natural Numbers
The set of positive integers (1, 2, 3, ...) that are used in counting. Factorials are computed using these numbers.
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Permutations
Permutations represent the number of different possible ways we can arrange a number of elements.
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Characteristics of Permutations
• Arranging all elements within the sample space.
• No repetition.
• Pn = n × (n − 1) × (n − 2) × ⋯ × 1 = n! (Called “n factorial”)
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In how many ways can we arrange 5 people?
120
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Variations
Variations represent the number of different possible ways we can pick and arrange a number of elements.
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Variations (With Repetition)
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Variations (Without Repetition)
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Combinations
Combinations represent the number of different possible ways we can pick a number of elements.
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Characteristics of Combinations
• Takes into account double-counting.
• All the different permutations of a single combination are different variations.
• Combinations are symmetric, so C(p, n) = C((n−p), n) , since selecting p elements is the same as omitting n-p elements.
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Combinations (Separate Sample Space)
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Characteristics of Combinations (Separate Sample Space)
• The option we choose for any element does not affect the number of options for the other elements.
• The order in which we pick the individual elements is arbitrary.
• We need to know the size of the sample space for each individual element.
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Combinations with Repetition
Combinations represent the number of different possible ways we can pick a number of elements. In special
cases we can have repetition in combinations and for those we use a different formula.
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Set
A set is a collection of elements, which hold certain values. Additionally, every event has a set of outcomes that satisfy it.
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The Null-Set
The null-set (or empty set), denoted ∅, is an set which contain no values.
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Finite Set
A set with a limited number of elements, which is a fundamental concept in combinatorics.
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𝑥 ∈ A
Element x is a part of set A.
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A ∋ 𝑥
Set A contains element x.
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𝑥 ∉ 𝐴
Element x is NOT a part of set A.
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∀𝑥:
For all/any x such that…
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A ⊆ B
A is a subset of B.
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Intersection (A ∩ B)
The intersection of two or more events expresses the set of outcomes that satisfy all the events simultaneously.
Graphically, this is the area where the sets intersect.
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Union (A ∪ B)
The union of two or more events expresses the set of outcomes that satisfy at least one of the events.
Graphically, this is the area that includes both sets.
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Union Formula
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Mutually Exclusive Sets
Sets with no overlapping elements (A ∩ B = ∅) are called mutually exclusive. Graphically, their circles never touch.
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Independent Events
If two events are independent, the probability of them occurring simultaneously equals the product of them occurring on their own.
P(A∩B) = P(A) × P(B).
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Dependent Events
Two or more events in which the outcome of one event affects the outcome of the other(s). Mathematically, events A and B are dependent if and only if: P(A∩B) ≠ P(A) × P(B).
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Conditional Probability
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P(A|B)
The probability of event A occurring given that event B has occurred.
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P(B|A)
The probability of event B occurring given that event A has occurred.
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Law of Total Probability
The law of total probability dictates that for any set A,
which is a union of many mutually exclusive sets B1,B2, … , Bn, its probability equals the following sum.
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Additive Law
The additive law calculates the probability of the union based on the probability of the individual sets it accounts for.
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The Multiplication Rule
The multiplication rule calculates the probability of the intersection based on the conditional probability.
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Bayes' Law
Bayes’ Law helps us understand the relationship between two events by computing the different conditional probabilities.
We also call it Bayes’ Rule or Bayes’ Theorem.
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Distribution
A distribution shows the possible values a random variable can take and how frequently they occur.
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Probability Function
A function that assigns a probability to each distinct outcome
in the sample space. P(Y = y), equivalent to p(y), where Y is the actual outcome and y is one of the possible outcomes.
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Discrete Distributions
• Have a finite number of outcomes.
• Can add up individual values to determine
probability of an interval.
• Expected Values might be unattainable.
• Graph consists of bars lined up one after the
other.
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Continuous Distributions
• Have infinitely many consecutive possible
values.
• Cannot add up the individual values that make
up an interval because there are infinitely
many of them.
• Graph consists of a smooth curve.
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Variance
A measure of how far a set of numbers are spread out from their average value.
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Standard Deviation
A measure of the amount of variation or dispersion of a set of values, calculated as the square root of the variance.
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Uniform Distribution (Y ~ U(a, b))
A distribution where all the outcomes are equally likely is called a Uniform Distribution.
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Bernoulli Distribution (Y ~ Bern(p))
A distribution consisting of a single trial and only two possible outcomes – success or failure is called a Bernoulli Distribution.
• E(Y) = p.
• Var(Y) = p × (1 − p).
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Binomial Distribution (Y~ B(n, p))
A sequence of identical Bernoulli events is called Binomial and follows a Binomial Distribution.
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Poisson Distribution (Y~ Po(λ))
When we want to know the likelihood of a certain event occurring over a given interval of time or distance, we use a Poisson Distribution.
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Euler’s Number (e)
A mathematical constant approximately equal to 2.72, used in various mathematical calculations, including the Poisson Distribution.
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A Normal Distribution represents a distribution that most natural events follow.
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Mean (μ)
The average value of a set of numbers, used as a parameter to define a Normal Distribution.
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A measure of how spread out the values in a data set are, used as a parameter to define a Normal Distribution.
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Symmetry in Normal Distribution
The property of a Normal Distribution where the left and right sides of the graph are mirror images of each other.
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Bell-Shaped Curve
The characteristic shape of the graph of a Normal Distribution.
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Outliers
Data points that fall far outside the majority of the other points in a data set.
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68-95-99.7 Rule
A rule that describes the percentage of data within 1, 2, and 3 standard deviations of the mean in a Normal Distribution (approximately 68%, 95%, and 99.7%, respectively).
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Standardizing a Normal Distribution
To standardize any normal distribution we need to transform it so that the mean is 0 and the variance and standard deviation are 1.
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Students’ T Distribution (Y~ t(k))
Represents a small sample size approximation of a Normal Distribution.
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• Its graph is asymmetric and skewed to the right.
• E(Y)= k.
• Var(Y) = 2k.
• The Chi-Squared distribution is the square of the t-distribution.
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Scale Parameter
The rate parameter of an exponential distribution that determines the shape of its probability distribution function (λ).
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Exponential Distribution (Y ~ Exp(λ))
The Exponential Distribution is usually observed in events which significantly change early on.
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Location Parameter
In the context of the logistic distribution, it represents the mean of the distribution and is synonymous with the location.
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Logistic Distribution (Y~ Logstic(μ, s))
The Continuous Logistic Distribution is observed when trying to determine how continuous variable inputs can affect the probability of a binary outcome.
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Correlation
A statistical measure that describes the extent to which two variables are related and change together.
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Categorical Outcomes
Outcomes that can be placed into distinct categories but do not have an inherent order or numerical value.
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Numerical Outcomes
Outcomes that are represented by numbers and can be placed in numerical order.
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Option Pricing
A fundamental concept in financial mathematics, option pricing involves calculating the fair value of options (financial derivatives), using models that incorporate various factors and probabilistic assumptions.
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Financial Derivatives
Financial securities whose value is dependent on or derived from an underlying asset or group of assets.
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Risk Assessment
The process of identifying, analyzing, and accepting or mitigating uncertainty in investment decisions.
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Uncertain Future Events
Refers to future occurrences in the financial world that cannot be predicted with certainty, necessitating the use of probability and statistical models to estimate outcomes.