Lecture 27: Conditional Expectation given an R.V. | Statistics 110

TL;DR

Conditional expectation is the expected value of a random variable given the value of another variable. It is calculated by considering specific conditions and can reveal how one variable relates to another. Key properties include that the expectation of the conditional expectation equals the unconditional expectation, a principle known as Adam's Law.

Transcript

All right. So let's get started. So we're still talking about conditional expectation, right? And so today we'll finish conditional expectation as a topic in its own right, which of course, doesn't mean you can then forget it because everything in this course is about thinking conditionally. But as its own topic, we'll finish that today. So I wante... Read More

Key Insights

• 👻 Conditional expectation allows for the prediction of a random variable based on certain conditions or information.
• ❓ Conditional expectation is a random variable itself and can be treated as a function of the conditioning variable.
• 💁 Conditional variance measures the spread or dispersion of a random variable when conditional information is known.
• 🟰 Adam's Law states that the expectation of the conditional expectation is equal to the unconditional expectation.

Questions & Answers

Q: What is conditional expectation and how is it different from unconditional expectation?

Conditional expectation is the expected value of a random variable given certain information or conditions. It represents the best prediction of a random variable when some information is known. Unconditional expectation, on the other hand, is the average or expected value of a random variable without any conditions or information.

Q: How can conditional expectation be used to make predictions?

Conditional expectation allows us to make predictions by considering the information or conditions that are known. By conditioning on certain variables or events, we can use conditional expectation to estimate the expected value or outcome of a random variable.

Q: What are the key properties of conditional expectation?

Some key properties of conditional expectation include taking out what's known, which allows us to extract a known variable from the conditional expectation, and the fact that the expectation of the conditional expectation is equal to the unconditional expectation. Additionally, conditional expectation is a random variable itself and is a function of the conditioning variable.

Q: How is conditional variance defined and how does it relate to conditional expectation?

Conditional variance is the measure of the spread or dispersion of a random variable given certain conditions or information. It is defined as the expected square difference from the conditional mean. It relates to conditional expectation by allowing us to measure the variability of a random variable when conditional information is known.

Summary

In this video, we finished discussing conditional expectation as a topic on its own. We started with some simple examples to understand the concept and notation of conditional expectation. Then, we derived some properties of conditional expectation, such as taking out what's known and the relationship between conditional expectation and independence. Additionally, we introduced iterated expectation, which is a generalization of the law of total probability. We also discussed the concept of conditional variance and proved Eve's Law, which states that the variance of a random variable is equal to the expected value of the conditional variance plus the variance of the conditional expectation.

Questions & Answers

Q: What are some properties of conditional expectation?

Some properties of conditional expectation include taking out what's known, the relationship between conditional expectation and independence, iterated expectation, and Eve's Law.

Q: What is the concept of taking out what's known?

Taking out what's known refers to the ability to factor out any function of a random variable that is treated as a constant when conditioning on another random variable.

Q: How is conditional expectation related to independence?

If two random variables X and Y are independent, then the conditional expectation of Y given X is equal to the unconditional expectation of Y.

Q: What is the concept of iterated expectation?

Iterated expectation, also known as Adam's Law, allows us to compute the expected value of a random variable by breaking it down into conditional expectations, simplifying the calculation.

Q: What is Eve's Law?

Eve's Law states that the variance of a random variable is equal to the expected value of the conditional variance plus the variance of the conditional expectation.

Q: How can we use conditional expectation to find the mean and variance of a random variable?

By conditioning on relevant variables, we can compute the conditional expectation of the random variable and then apply Eve's Law to obtain the mean and variance.

Q: What is the intuition behind Adam's Law and Eve's Law?

Adam's Law and Eve's Law help us simplify calculations by conditioning on relevant information and breaking down the problem into conditional expectations and variances.

Q: How can we find the conditional expectation and variance of a random variable given a beta distribution of a parameter?

To find the conditional expectation and variance, we can use LOTUS (Law of the Unconscious Statistician) to evaluate the expected value and variance of the random variable given the beta distribution.

Q: Why is Beta distribution commonly used in modeling the distribution of a parameter?

The Beta distribution is a versatile family of distributions that takes continuous values between 0 and 1, making it a suitable choice for modeling proportions or probabilities. Additionally, it is a conjugate prior for the binomial distribution, which has mathematical conveniences and is widely used in practice.

Q: What are the main takeaways from this video?

The main takeaways include understanding the properties of conditional expectation, such as taking out what's known, the relationship with independence, iterated expectation, and Eve's Law. These concepts are useful tools for simplifying calculations and understanding conditional distributions.

Summary & Key Takeaways

• The content explores conditional expectation by providing examples of conditioning on a random variable and deriving properties such as taking out what's known and the symmetry of conditional distributions.

• It also introduces the concept of conditional variance and its definition as the expected square difference from the conditional mean.

• The video then discusses key insights such as Adam's Law, which states that the expectation of the conditional expectation is equal to the unconditional expectation, and Eve's Law, which states that the variance of a random variable can be decomposed into the expected value of the conditional variance plus the variance of the conditional expectation.