https://r2.erweima.ai/mathbot/image/17265309593208b1mfele.jpg

Let's solve **Question 2** from the image.

The probability density function (PDF) is given by:

\[
f(x) = \begin{cases}
\frac{1}{8} + \frac{3}{8}x & \text{for } 0 \leq x \leq 2 \\
0 & \text{otherwise}
\end{cases}
\]

We are asked to find the probability that the dynamic load \(X\) is less than \(0.8\) N, i.e., \(P(X < 0.8)\).

### Step-by-step solution:
To find the probability, we need to integrate the PDF over the interval from 0 to 0.8:

\[
P(X < 0.8) = \int_0^{0.8} \left( \frac{1}{8} + \frac{3}{8}x \right) dx
\]

Now, let's compute this integral. I'll do the calculation.The probability that the dynamic load \( X \) is less than 0.8 N is approximately \( 0.2200 \).

Would you like further details or explanations?

Here are five related questions you could explore:
1. How is a probability density function (PDF) related to cumulative distribution functions (CDF)?
2. What would the probability be for \( X \) being greater than 0.8 N?
3. How do you normalize a probability density function to ensure it's valid?
4. What is the total probability that \( X \) lies between 0 and 2 N?
5. Can the same method be applied to find the probability of \( X \) being in any given interval?

**Tip:** Always ensure the total area under a probability density function equals 1.

Suponga que la función de densidad de probabilidad de la magnitud X de una carga dinámica sobre un puente (en newtons) está dada por: f(x) = (1/8 + 3/8x) para 0 ≤ x ≤ 2, de lo contrario f(x) = 0. Calcule la probabilidad de que la carga dinámica sea menor que 0.8 N.

Learn how to calculate the probability that a dynamic load on a bridge is less than 0.8 N, using the probability density function f(x) = (1/8 + 3/8x) for 0 ≤ x ≤ 2. This step-by-step solution involves integration and key concepts of probability theory, suitable for college-level students.

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