Is My Formula for Conditional Expectation Correct?

In summary, the conversation is about finding the conditional expectation of a random variable Y, given that it is less than another independent random variable Z. The formula for this is E(Y|Y<z) = (integral of y*f(y) from 0 to z) / F(z), where f(y) is the pdf of Y and F(z) is the cdf of Z. The individual asking the question is unsure if this formula is correct, but it is confirmed by the other person in the conversation. The reasoning behind the formula is also discussed.
  • #1
JamesF
14
0
This result isn't in our book, but it is in my notes and I want to make sure it's correct. Please verify if you can.

Homework Statement


I have two I.I.D random variables. I want the conditional expectation of Y given Y is less than some other independent random variable Z.

[tex] E(Y \, \vert \, Y < z) = \dfrac{\int_0^{z} y \cdot f(y) \, dy}{F(z)} [/tex]

Where f(y) is the pdf of Y and F(z) is the cdf for Z

The Attempt at a Solution


I've searched the book and the web, but all I find is the formula for conditional expectation for [tex] E(X | Y = y) [/tex] for joint distributions and the like. Is my formula correct?
 
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  • #2
You know that [tex]\mathbb{E}[X|Y]=\frac{\mathbb{E}[X \mathbf{1}_Y]}{\mathbb{P}(Y)}[/tex] so your formula looks correct.
 
Last edited:
  • #3
Think this way: if you know [tex] Y \le z[/tex], then the truncated distribution has density

[tex]
g(y \mid Y \le z) = \frac{f(y)}{F(z)}
[/tex]

so the expectation is

[tex]
\int_0^z y g(y \mid Y \le z) \, dy = \frac{\int_0^z y f(y) \, dy}{F(z)}
[/tex]

exactly as you have it.
 

What is conditional expectation?

Conditional expectation is a statistical concept that calculates the expected value of one random variable given the value of another random variable. It represents the average outcome of a variable if it is known that another variable has a specific value.

How is conditional expectation calculated?

Conditional expectation is calculated using the formula E(Y|X) = ∫ Y f(y|x) dy, where Y is the random variable of interest, X is the conditioning variable, and f(y|x) is the conditional probability density function.

What is the difference between conditional expectation and unconditional expectation?

Unconditional expectation is the average value of a random variable without any conditions, while conditional expectation takes into account a specific condition or value of another variable. In other words, conditional expectation is a more precise measure of the average outcome.

What is the significance of conditional expectation in statistics?

Conditional expectation is important in statistics because it allows us to make predictions about one variable based on the value of another variable. It also helps to understand the relationship between two variables and how they affect each other.

In what fields is conditional expectation commonly used?

Conditional expectation is used in a variety of fields, including economics, finance, and engineering. It is also commonly used in probability theory and statistics to analyze and model random phenomena. It has applications in decision-making, risk assessment, and forecasting.

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