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Probability Theory Fundamentals

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Central Limit Theorem

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States that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution. Example: The average height of randomly selected people will be normally distributed if the sample size is large.

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Random Variable

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A variable whose possible values are numerical outcomes of a random phenomenon. Example: The sum of the roll of two dice.

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Probability Density Function (PDF)

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A function that describes the relative likelihood for a continuous random variable to take on a given value. Example: The normal distribution curve is a PDF.

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Discrete Uniform Distribution

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A probability distribution where each of the n finite outcomes has equal probability 1/n. Example: Rolling a fair six-sided dice.

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Independent Events

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Two events are independent if the occurrence of one does not affect the probability of the other. Example: Tossing two different coins.

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Sample Space

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The set of all possible outcomes of a probabilistic experiment. Example: For a coin toss, the sample space is {Heads, Tails}.

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Variance

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A measure of the spread of a distribution about the expected value. Given by Var(X)=E[(XE(X))2]Var(X) = E[(X - E(X))^2]. Example: Variance of a dice roll is about 2.92.

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Standard Deviation

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The square root of the variance, a measure of the spread of a distribution. Example: Standard deviation of a dice roll is about 1.71.

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Law of Large Numbers

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States that as the number of trials increases, the sample average will converge to the expected value. Example: Flipping a coin many times will get close to an average of 0.5 heads.

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Poisson Distribution

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A discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. Given by P(X=k)=eλλkk!P(X=k) = \frac{{e^{-\lambda} \lambda^{k}}}{k!} for k=0,1,2,...k = 0,1,2,... Example: Number of emails received in an hour.

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Probability of an Event

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A measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1. Example: Probability of getting Heads in a coin toss is 0.5.

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Expected Value

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The theoretical average of a random variable over many trials. Calculated as E(X)=xP(x)E(X) = \sum xP(x) for a discrete variable. Example: Expected value of a dice roll is 3.5.

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Binomial Distribution

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A probability distribution that summarizes the likelihood that a value will take one of two independent states across a number of observations or trials. Given by P(X=k)=(nk)pk(1p)nkP(X=k) = \binom{n}{k} p^k(1-p)^{n-k} for k=0,1,2,...,nk = 0,1,2,...,n. Example: Number of heads in 10 coin tosses.

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Bayes' Theorem

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A formula that describes how to update the probabilities of hypotheses when given evidence. Given by P(AB)=P(BA)P(A)P(B)P(A|B) = \frac{P(B|A)P(A)}{P(B)}. Example: Revising the probability of having a disease, given a positive test result.

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Probability Mass Function (PMF)

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A function that gives the probability that a discrete random variable is exactly equal to some value. Example: PMF of a fair dice roll giving the probability of each number 1 through 6 is 1/6.

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Correlation Coefficient

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A normalized measure of the strength and direction of the relationship between two random variables. Given by ρ=Cov(X,Y)σXσY\rho = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}. Example: Correlation between study time and exam score.

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Cumulative Distribution Function (CDF)

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A function that gives the probability that a random variable is less than or equal to a certain value. Example: CDF of a standard normal distribution at z=0 is 0.5.

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Event

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A subset of the sample space to which a probability is assigned. Example: Getting a Head in a coin toss is an event.

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Conditional Probability

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The probability of an event given that another event has occurred. Given by P(AB)=P(AB)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}, if P(B)>0P(B) > 0. Example: Probability of drawing an ace from a deck of cards, given that a heart has been drawn.

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Covariance

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A measure of the extent to which two random variables change together. Given by Cov(X,Y)=E[(XE(X))(YE(Y))]Cov(X,Y) = E[(X - E(X))(Y - E(Y))]. Example: Covariance between height and weight.

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