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**SUMMARY:**

Let's define N variables called X, Y, Z, .... in probability space. The joint probability distribution is finding the probability of each variable that falls in a certain defined range or discrete set of values for that particular variable. Now when there are only two variables in the probability space, then it is known as bivariate distribution.

**DEFINITION:**

It is the probability that a specific event will occur when there are two independent random variables in the scenario. For example, having two bowls, each filled with two different types of candies, and picking one candy from each bowl gives two independent random variables, for the two different candies. Since we are picking one candy from each bowl at the same time, you have a bivariate distribution when calculating your probability of ending up with particular kinds of sweets.

Bivariate probability distributions is about the relationship between two variables. This will tell whether or not two variables are independent. Then also we want to know how two variables are related if they are not independent, e.g. if income & education are not independent, do people with more knowledge tend to have higher incomes or lower incomes? This type of problems can be solved with the help of bivariate distribution.

**BIVARIATE NORMAL DISTRIBUTION:**

If two variables in the bivariate distribution usually are distributed then this bivariate distribution is known as bivariate normal distribution. The fact is that both variables are typically allocated that doesn't imply that they also have a joint normal distribution.

This type of distribution is a particular case of a multivariate normal distribution which has a broad application in real time solutions. This type of distribution has heavy applications in biology and economy where the relationship between variables is of more significant interest.

For instance: We toss a pair of four-sided dice in which one of the dice is White, and one of the dice is Black, where

X = the outcome of the white dice = {1, 2, 3 or 4}

Y = the outcome of the black dice = {1, 2, 3 or 4}

What is the probability of X takes on particular value x and Y takes on particular value y? This is a case of bivariate distribution. Here we have to find P (X=x, Y=y).

Solution: Now here in case of two dices we have to find "joint probability distribution" of X and Y for which we first have to define the support of X and Y. The support of X is:

S1 = {1, 2, 3, 4}

And, the support of Y is:

S2 = {1, 2, 3, 4}

Now, if we let (x, y) denote one of the possible outcomes of one toss of the pair of dice, then certainly (1, 1) is a possible outcome, as is (1, 2), (1, 3) and (1, 4). If we continue to list all of the possible outcomes, we soon see that the joint support S has 16 possible outcomes:

S = {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)}

As the dice are fair, we will expect each of the 16 possible outcomes to be equal. Therefore, let us use the classical approach of assigning a probability, the probability that X equals to any specific x value, & Y equals any specific y value, is 1/16. I.e., for all (x, y) in the support S:

P (X = x, Y = y) = 1/16

Thus, bivariate distributions are used heavily where we have to define relations between two variables and to find out the effect of one variable over other.

Let's take one more example.

Suppose, we have to find out individuals with obesity in the large data set. Take one variable Y which is the weight of a randomly selected individual, in a pound. Now we are interested in finding the probability of a randomly selected individual weight between 140 and 160 pounds. Now if think about it, we will get an idea that the weight of an individual increase as height increases.

So, now we might have an interest in finding the weight that is between 140 and 160 pounds with a given specific height. At this point, our bivariate distribution takes place in the picture.

As we conclude here,

- We can say that in any real-life scenario or real time calculation requires co-relation between two variables, which is obtained by bivariate distribution.
- If there are more than two variables included, then it is known as multivariate distribution.
- If both variables behave in a normal distribution pattern, then it will be remembered as the bivariate normal distribution. It does not imply that both variables will show joint normal distribution.
- Bivariate distribution is used to find the probability of occurring an event based on the occurrence of another event.

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