Can you solve conditional probability with the laws of covariance?
These conditional probability questions can seem mysterious at first, but with a solid grip on the Laws of Total Expectation, Variance, and Covariance we can solve them easily and efficiently.
How are the laws of total expectation used to solve conditional probability problems?
In this article, we’ll see how to use the Laws of Total Expectation, Variance, and Covariance, to solve conditional probability problems, such as those you might encounter in a job interview or while modeling business problems where random variables are conditional on other random variables.
How to calculate the law of total variance?
Since we are calculating the variance, there are 2 sources of variability: (expected within the group variability in A2) + (variability in the expected value of A2 across the groups). To understand this better, have a look at this formula: This explains the intuition behind the Law of Total Variance very clearly, which is summarised here:
How to calculate the conditional variance of L1?
Let’s start by calculating the variance of L1, denoted by Var (L1). Since L1 is not dependent on any other variable, we can solve for Var (L1) directly by using the basic formula. Now let’s look into the variance for A2. Again, since A2 is dependent on L1, taking the conditional variance makes the calculation easier.
Which is the conditional covariance matrix of Y?
The conditional variance-covariance matrix of Y given that X = x is equal to the variance-covariance matrix for Y minus the term that involves the covariances between X and Y and the variance-covariance matrix for X. For now we will call this conditional variance-covariance matrix A as shown below:
How are unconditional and conditional variances collected in a matrix?
Just as the unconditional variances and covariances can be collected into a variance-covariance matrix Σ, the conditional variances and covariances can be collected into a conditional variance-covariance matrix: Note!
Which is the covariance between Y I and Y J?
This can be interpreted as the covariance between Y i and Y j given a sample from the subpopulation where X = x. Just as the unconditional variances and covariances can be collected into a variance-covariance matrix Σ, the conditional variances and covariances can be collected into a conditional variance-covariance matrix: