The image shows the integral:

$\int_{0}^{\infty} y f_Y(y) \, dy$

This expression represents the expected value (mean) of a continuous random variable $Y$ with probability density function $f_Y(y)$. In probability theory, the expected value of $Y$ is defined as:

$E[Y] = \int_{0}^{\infty} y f_Y(y) \, dy$

where:

• $f_Y(y)$ is the probability density function (PDF) of the random variable $Y$,
• $y$ is the variable of integration,
• The limits of the integral (from 0 to $\infty$) suggest that $Y$ is a non-negative continuous random variable.

This integral computes the weighted average of $y$, where the weights are given by the PDF $f_Y(y)$.

Would you like more detailed information about expected value or related concepts?

Here are 5 related questions:

1. How is the expected value of a discrete random variable calculated?
2. What properties of the expected value make it useful in probability theory?
3. Can you explain how to find $f_Y(y)$ if the cumulative distribution function (CDF) is known?
4. What are common continuous distributions where this type of integral is applied (e.g., exponential, normal)?
5. How is variance related to the expected value?

Tip: The expected value of a function of a random variable, such as $E[g(Y)]$, can be calculated using a similar integral: $E[g(Y)] = \int_{0}^{\infty} g(y) f_Y(y) \, dy$.