Probability theory is the branch of mathematics concerned with analyzing random phenomena and modeling uncertainty. It provides a framework for quantifying and reasoning about the likelihood of events.<\/p>\n\n\n\n
A probability space is a mathematical construct that models a random experiment. It consists of three components:<\/p>\n\n\n\n
The probability measure \\( P \\) must satisfy the following axioms:<\/p>\n\n\n\n
The conditional probability of an event \\( A \\) given that \\( B \\) has occurred is defined as:
$$ P(A|B) = \\frac{P(A \\cap B)}{P(B)} $$<\/p>\n\n\n\n
Bayes’ Theorem relates the conditional probabilities of two events:
$$ P(A|B) = \\frac{P(B|A)P(A)}{P(B)} $$<\/p>\n\n\n\n
A random variable is a function that assigns a real number to each outcome in the sample space. There are two types of random variables:<\/p>\n\n\n\n
The probability distribution of a random variable describes how probabilities are distributed over its possible values.<\/p>\n\n\n\n
For a discrete random variable \\( X \\), the probability mass function (PMF) \\( p(x) \\) gives the probability that \\( X \\) takes the value \\( x \\):
$$ p(x) = P(X = x) $$<\/p>\n\n\n\n
For a continuous random variable \\( X \\), the probability density function (PDF) \\( f(x) \\) describes the relative likelihood of \\( X \\) taking on a particular value:
$$ P(a \\leq X \\leq b) = \\int_a^b f(x) \\, dx $$<\/p>\n\n\n\n
The expectation (or mean) of a random variable \\( X \\) is a measure of its central tendency. For a discrete random variable:
$$ \\mathbb{E}[X] = \\sum_{x} x p(x) $$<\/p>\n\n\n\n
For a continuous random variable:
$$ \\mathbb{E}[X] = \\int_{-\\infty}^{\\infty} x f(x) \\, dx $$<\/p>\n\n\n\n
The variance of a random variable \\( X \\) measures the spread of its values. It is defined as:
$$ \\text{Var}(X) = \\mathbb{E}[(X – \\mathbb{E}[X])^2] $$<\/p>\n\n\n\n
Using the linearity of expectation:
$$ \\text{Var}(X) = \\mathbb{E}[X^2] – (\\mathbb{E}[X])^2 $$<\/p>\n\n\n\n
A discrete distribution representing the number of successes in a fixed number of independent Bernoulli trials:
$$ P(X = k) = \\frac{n!}{k!(n-k)!} p^k (1-p)^{n-k} $$<\/p>\n\n\n\n
A continuous distribution characterized by its mean \\( \\mu \\) and variance \\( \\sigma^2 \\):
$$ f(x) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} e^{-\\frac{(x – \\mu)^2}{2 \\sigma^2}} $$<\/p>\n\n\n\n
A continuous distribution describing the time between events in a Poisson process:
$$ f(x) = \\lambda e^{-\\lambda x} $$<\/p>\n\n\n\n
The law of large numbers states that as the number of trials increases, the sample mean converges to the expected value:
$$ \\frac{1}{n} \\sum_{i=1}^{n} X_i \\rightarrow \\mathbb{E}[X] \\quad \\text{as} \\quad n \\to \\infty $$<\/p>\n\n\n\n
The central limit theorem states that the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution, regardless of the original distribution:
$$ \\frac{\\sum_{i=1}^{n} X_i – n \\mu}{\\sigma \\sqrt{n}} \\rightarrow N(0, 1) \\quad \\text{as} \\quad n \\to \\infty $$<\/p>\n\n\n\n
Probability theory provides a rigorous foundation for analyzing random phenomena and making inferences about uncertain events. Its concepts and methods are essential for understanding and modeling a wide range of real-world processes.<\/p>\n","protected":false},"excerpt":{"rendered":"
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