Expectation and Distribution(MRV)

Types of Random Variables:

There are three types of random variables:

• Discrete random variables
• Continuous random variables
• Mixed random variables

Expected Value of Multivariate Random Variables

As the name suggests, expected values are the predicted probability distribution of the multivariate random variable. Let us take an example:

If a quiz contains 20 questions multiple-choice with A, B, C, D as the answers, and you mark all “A,” then you can expect to get 25% right. 

The maths behind this kind of expected value is:
The probability P of getting each question correct=¼= 0.25
The number of questions in the quiz (n)*= 20
P x n = 0.25 x 20 = 5

Mathematical formula

The mathematical formula for calculating the expected value of the multivariate random variable is the probability of an event occurring multiplied by the number of times the event occurred.

P(x)*n

Where,
P(x) is the probability of an event occurring
N is how many times the event occurred

Source: Link

Distribution of Multivariate Random Variable

The probability distribution of a multivariate random variable  X  is a project of probabilities to periods of decimal numbers the use of a feature f(x), referred to as a density function, in the following manner: the chance that  X  assumes a value in the range  [a,b]  is equal to the place of the region this is bounded above through the graph of the equation  y=f(x), bounded under the x-axis, and bounded at the left and right through the vertical lines through a  and b.

Source: Link

Covariance of Multivariate Random variables

The covariance of multivariate random variables is responsible for finding the dependency between pairs of random variables. For random variables X1 and X2, their covariance is defined by the mathematical formula:

Source: Link

Expected Value of Bivariate Random Variable

The expected value in the bivariate random variable is predicted by assuming only the probability of the first outcome as success and the rest as a failure.
For example, if you research a group of college students to find out their average GATE score and their age, you have two pieces of the puzzle to find (GATE score and age).

Mathematical formula

In Bivariate Random Variable, the mathematical representation is 

Source: Link

Distribution of Bivariate Random Variable

To gain the marginal distribution over a subset of multivariate random variables, one only wishes to drop the inappropriate variables (the variables that one desires to marginalize out) from the mean vector and the covariance matrix.

Mathematical formula

Source: Link

Source: Link

Applications of Multiple random variables

• Find out the number of car accidents
• Number of customers in a day/month/year
• To specify a joint distribution via conditional and marginal distributions

FAQs

1. What is a Random variable?
   A Random variable is the set of probable values of the random experiment.
    
2. How many types of Random variables are there?
   There are three types of Random variables; Discrete, Continuous, and Mixed.
    
3. How do you identify a random variable?
   Random variables are generally represented by capital letters, for example, X and Y.

Key Takeaways

In this article, we learned about multivariate and bivariate random variables, their expected values, and distribution. To get in-depth knowledge of machine learning, check out our industry-level courses on coding ninjas.