https://file.aiquickdraw.com/mathbot/file/1725772412276zqk4hg8e.jpg

The text extraction from the image didn't work well. Let me instead describe and interpret the image content directly.

From what I can see in the image, it appears that we are dealing with a problem involving a probability density function (PDF), particularly in the context of statistics. The problem asks for two things:

1. **Determine the value of \( K \)** (a constant).
2. **Calculate the mean \( E(X) \)**.

The provided probability density function (PDF) is:
\[
f(x) = K \cdot e^{-\frac{(x - 3)^2}{2}}, \text{ for } 0 \leq x \leq 6.
\]

This appears to be a Gaussian-like (normal distribution) PDF, with \( K \) being a normalizing constant. I will walk through the steps necessary to determine \( K \) and calculate \( E(X) \) based on this interpretation.

Would you like to proceed with solving these two questions?

Given a probability density function f(x) = K * e^(-(x-3)^2 / 2) for 0 <= x <= 6, determine the value of K and calculate the mean E(X).

This problem involves finding the normalization constant K and calculating the mean E(X) for a Gaussian-like probability density function, f(x) = K * e^(-(x-3)^2 / 2) within the range 0 <= x <= 6. Explore the concepts of PDF normalization and continuous random variable mean calculation.

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