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.