https://file.aiquickdraw.com/mathbot/file/1725799410589cr57v5st.jpg

The image you uploaded shows a mathematical expression involving a probability statement and a double integral:

\[
P(X < \frac{1}{2} \text{ and } Y < \frac{1}{2}) = \int_0^{\frac{1}{2}} \int_0^{\frac{1}{2}} f(x, y) \, dx \, dy
\]

This represents the probability that two continuous random variables \(X\) and \(Y\) are both less than \(1/2\), given their joint probability density function \(f(x, y)\).

To break it down:

1. The left side, \(P(X < \frac{1}{2} \text{ and } Y < \frac{1}{2})\), is the probability that both \(X\) and \(Y\) are less than 1/2.
   
2. The right side shows the calculation of this probability using a double integral:
   \[
   \int_0^{\frac{1}{2}} \int_0^{\frac{1}{2}} f(x, y) \, dx \, dy
   \]
   where \(f(x, y)\) is the joint probability density function of \(X\) and \(Y\). This integral computes the total probability over the region where both \(X\) and \(Y\) lie between 0 and 1/2.

Would you like further details or have any questions about the interpretation? Here are some related questions:

1. What is a joint probability density function?
2. How do you evaluate a double integral in probability theory?
3. What are the conditions for \(f(x, y)\) to be a valid joint density function?
4. Can this integral be computed directly for specific forms of \(f(x, y)\)?
5. How does this formula extend to other bounds or regions for \(X\) and \(Y\)?

**Tip:** When working with joint probabilities, always ensure that the limits of integration correctly define the region of interest.

This problem demonstrates how to calculate the probability that two continuous random variables X and Y are both less than 1/2 using a double integral. Key concepts include probability theory, joint probability density functions, and double integrals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation