Final answer:
A bivariate probability distribution describes the probability of outcomes based on two variables. Binomial and uniform distributions are examples of bivariate and continuous probability distributions respectively. Marginal distributions, another important concept, reflect category totals for one variable, considering another.
Step-by-step explanation:
A bivariate probability distribution is a statistical distribution that describes the probability of outcomes of two random variables. It's another name for a two-variable data distribution. The bivariate distribution assesses the probability that a certain event will occur based on two variables. For example, when analyzing bivariate data, we often come across two-way frequency tables, and the sums in the margins - hence referred to as marginal frequencies or marginal distributions. Marginal distributions require bivariate data since they reflect totals for each category across one variable while considering the other variable.
Consider the binomial probability distribution, which is a discrete random variable that arises from Bernoulli trials. Here, we have a fixed number (n) of independent trials, and 'p' represents the probability of success on each trial. The notation X ~ B(n, p) indicates a random variable X that follows a binomial distribution with parameters n and p.
A uniform distribution is another type of probability distribution for a continuous random variable with equally likely outcomes over a certain range, often described as rectangular because the graph of the probability distribution function is shaped like a rectangle. For example, the probability for the domain 'a < x < b' is uniformly distributed and the cumulative distribution can be represented as 'P(X ≤ x) = (x - a) / (b - a)' where 'a' and 'b' define the range of the distribution.