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probability
Fundamental concepts of probability

probability axioms

probability distributions

conditional probability

and Bayes' theorem
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They provide a
rigorous mathematical framework for understanding probability
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That is, for any event A, P(A) >= 0
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That is, P(S) = 1, where S is the sample space
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}, the probability of their
union is the sum of their individual probabilities
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) = P(A1) + P(A2) +
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Probability Distributions
A probability distribution is a function that describes the likelihood of different outcomes in a random
event
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The probability
distribution of a discrete random variable X is given by the probability mass function (PMF) p(x), which is
defined as:

p(x) = P(X = x)

where X is the random variable, and x is the outcome
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That is, Sum(p(x)) = 1
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2 Continuous Probability Distributions

Continuous probability distributions describe the probabilities of continuous events, such as the time it
takes for a customer to be served or the height of a randomly chosen person
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That is:

P(a <= X <= b) = Integral(f(x)dx) from a to b

The mean or expected value of a continuous random variable X is given by:

E[X] = Integral(x*f(x)dx) from -inf to +inf

The variance of a continuous random variable X is given by:

Var(X) = E[X^2] - (E[X])^2

Conditional Probability
Conditional probability is the probability of an event A given that another event B has occurred
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Bayes' Theorem
Bayes' theorem is a formula used to calculate conditional probabilities
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Bayes' theorem is used in
many fields, including statistics, machine learning, and artificial intelligence
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Independent Events
Two events A and B are said to be independent if the occurrence of one does not affect the occurrence
of the other
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If two events are independent, then the
joint probability of both events occurring is the product of their individual probabilities
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Law of Total Probability
The law of total probability is a theorem that states that the probability of an event A can be calculated
as the sum of the probabilities of A given each possible outcome of a second event B, weighted by the
probability of each outcome of B
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Expectation and Variance of a Random Variable
The expected value or mean of a random variable X is a measure of the central tendency of its
probability distribution
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It is
defined as:
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Var(X) = E[(X - E[X])^2]

where E[X] is the expected value of X
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This is true regardless of
the distribution of the individual random variables, as long as the sample size is sufficiently large
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The key concepts of probability covered in these study notes include
probability axioms, probability distributions, conditional probability, Bayes' theorem, independent
events, the law of total probability, expectation and variance of a random variable, and the central limit
theorem
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