Discover published sets by community

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

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

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.

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

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

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

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.

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.

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.

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

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

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.

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

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

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.

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.

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

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

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 $P(X=k) = \frac{{e^{-\lambda} \lambda^{k}}}{k!}$ for $k = 0,1,2,...$ Example: Number of emails received in an hour.