Law of Total Expectation (SOA Exam P - Probability - Multivariate Random Variables)

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After completing this video you should be able to:
- Calculate moments for joint, conditional, and marginal discrete random variables.
- Calculate variance and standard deviation for conditional and marginal probability distributions for discrete random variables only.
- Calculate joint moments, such as the covariance and the correlation coefficient for discrete random variables only.
The Law of Total Expectation, also known as the Law of Iterated Expectation or the Tower Property, is a fundamental concept in probability theory and statistics. It states that the expected value of a random variable can be computed by taking the conditional expectation of that variable given another random variable and then taking the expectation of the resulting conditional expectation.
Mathematically, if X and Y are two random variables, then the Law of Total Expectation can be expressed as:
E[X] = E[E[X|Y]]
This law allows us to break down complex expectations into simpler conditional expectations, providing a method for calculating the overall expected value based on conditional probabilities. It is particularly useful when dealing with situations involving multiple sources of uncertainty or when we need to estimate the average value of a random variable based on partial information.

[@joshuasonnen5982]

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[@Alexanderp488]

At 7:54 where did you get the values for E(x|y)?

[@liuliuzhang8359]

It is the expected value of X given Y=0,1,2, you can see at 5:33, the red part in the brackets after the calculation, it is actually the first row shows in 7:54 which is 22/25. Hope this can help you.