Discover published sets by community

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 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.

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.

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.

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.

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

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.

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.

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

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 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 $P(X=k) = \binom{n}{k} p^k(1-p)^{n-k}$ for $k = 0,1,2,...,n$. Example: Number of heads in 10 coin tosses.

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.

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

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.

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.