Bivariate Distribution

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A bivariate distribution gives probabilities for outcomes of two random variables occurring at the same time.

More precisely, it is a discrete joint distribution with two variables of interest, usually denoted X and Y. A “discrete joint distribution” has a finite set of possible outcomes (for example, 2, 3, or 99), whereas for the continuous version — the continuous joint distribution — you can describe relationships between an infinite number of variables.

Bivariate Distribution Table

Each bivariate distribution table is unique; its look depends on what variables you are studying. A table of rows and columns can be used to display discrete random variables with a finite (fixed) number of values. The rows represent one of the variables, the columns the other. For example, the following table shows the bivariate distribution of getting tails if you flip a coin three times:

bivariate distribution
Rows (X) = total number of coin flips, Columns (Y) = number of flips until a tail appears (if no tails appear in the trial, this is set to 0).
  • The table intersections represent an X-Y combination; the probabilities in those cells represent the joint probabilities.
  • The probabilities in all the cells add up to 1 (because this is a probability distribution, all the probabilities must add up to 1).
  •  Add probabilities across rows to get the probability distribution of random variable X (also called the marginal distribution of X).
  • Add probabilities down columns to get the probability distribution of random variable Y (also called the marginal distribution of Y). 

Example of working with the bivariate distribution.

Real Life Uses

Combining two variables in a single distribution, bivariate distributions are quite common both within and outside of the world of mathematics. For example, in medical checkups, cholesterol and triglyceride levels can be combined to assess heart health; gamblers may look at pairing sixes when rolling dice; cafés could try correlating coffee sales with those for cakes; while researchers investigate how alcohol plays into car crashes. Depending on which variables you choose to study however, each bivariate distribution will present unique insights – no unified representation is possible!

Bivariate Normal Distribution

A bivariate normal distribution describes the joint probability distribution of two normally distributed variables (X, Y). Five parameters describe this distribution:

  • mx, my = mean values of X and Y;
  • sx, sy = standard deviation of X and Y;
  • rxy = correlation coefficient between X and y.

On requirement for bivariate normal distributions is that zero correlation (r=0) means that X and Y are independent random variables.

Next: Marshall-Olkin Bivariate Distribution