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The problem in the image asks to compute probabilities based on a continuous probability density function (PDF) given by \( y = 0.5x + 0.2 \), for \( -0.4 \leq x \leq 1.6 \). The graph shows a line representing this function.

Let’s break down the tasks:

1. **The first part** asks for \( P(X < 1.2) \).
   - To find this probability, you need to integrate the PDF from the lower bound of \( x \) (i.e., \( -0.4 \)) to 1.2. This will give the area under the PDF curve, which represents the probability.

2. **The second part** asks for \( P(X > 1.2) \).
   - This is equal to \( 1 - P(X \leq 1.2) \), which means you subtract the probability found in the first part from 1.

I’ll guide you through solving these:

### Step 1: Find the normalization constant
Since this is a continuous PDF, the total area under the curve from \( x = -0.4 \) to \( x = 1.6 \) must equal 1. We can integrate the given PDF to check and normalize if necessary.

\[
\int_{-0.4}^{1.6} (0.5x + 0.2) \, dx
\]

### Step 2: Calculate \( P(X < 1.2) \)
To find \( P(X < 1.2) \), we will integrate the PDF from \( x = -0.4 \) to \( x = 1.2 \):

\[
P(X < 1.2) = \int_{-0.4}^{1.2} (0.5x + 0.2) \, dx
\]

### Step 3: Calculate \( P(X > 1.2) \)
This can be computed as:

\[
P(X > 1.2) = 1 - P(X \leq 1.2)
\]

Let me solve these for you using integration.The total area under the probability density function (PDF) from \( x = -0.4 \) to \( x = 1.6 \) is 1, confirming that the PDF is correctly normalized.

Here are the results for the probabilities:

1. \( P(X < 1.2) = 0.64 \)
2. \( P(X > 1.2) = 0.36 \)

These values represent the areas under the PDF curve for the respective ranges.

Do you want more details or have any questions?

Here are some related questions for further practice:
1. How would you compute \( P(0 < X < 1.2) \)?
2. What is the cumulative distribution function (CDF) for this PDF?
3. How would you find the median value of \( X \) given this PDF?
4. Can you determine the expected value (mean) of \( X \) based on the PDF?
5. How would the probabilities change if the PDF were modified to \( y = 0.5x + 0.3 \)?

**Tip:** The area under the PDF curve between any two points represents the probability that the random variable falls within that range.

Consider the continuous probability density function defined by y = 0.5x + 0.2 on -0.4 ≤ x ≤ 1.6. Compute the following probabilities: P(X < 1.2) = 0.64 and P(X > 1.2).

Learn how to calculate the probabilities P(X < 1.2) and P(X > 1.2) for a continuous probability density function given by y = 0.5x + 0.2 on the interval [-0.4, 1.6]. Detailed step-by-step solution with integration. Suitable for college-level or advanced high school students.

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