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.