The image you've provided appears to represent a probability density function (PDF), most likely for a continuous random variable. The shape in the plot resembles a rectangular (uniform) distribution with constant height. The graph suggests that the distribution is uniform over a range on the horizontal axis from 0 to 2, and the height of the graph seems to be 0.5.

To find the probability of being between 0.5 and 1.5, we can proceed as follows:

Steps:

1. Identify the total range: The uniform distribution seems to cover the range [0, 2].
2. Height of the rectangle: The height of the distribution (the probability density) appears to be 0.5.
3. Calculate the probability: The probability in a uniform distribution is given by the area under the curve over the desired interval.

For a uniform distribution, the probability of being within an interval $[a, b]$ is proportional to the length of the interval. Mathematically:

$P(0.5 \leq X \leq 1.5) = \text{Height} \times (\text{Length of the interval})$

• Height = 0.5 (as seen from the graph).
• Length of the interval = 1.5 - 0.5 = 1.

$P(0.5 \leq X \leq 1.5) = 0.5 \times 1 = 0.5.$

Conclusion:

The probability of the random variable being between 0.5 and 1.5 is 0.5.

Let me know if you need more details or further explanation! Here are five related questions to explore:

1. How would the probability change if the interval was from 0 to 1 instead of 0.5 to 1.5?
2. Can you explain why the uniform distribution is used in this case?
3. How is the area under the curve related to probability in continuous distributions?
4. What are the properties of a uniform probability distribution?
5. How can we calculate the expected value and variance of this uniform distribution?

Tip:

For any continuous probability distribution, the total area under the curve always equals 1, representing the total probability of all possible outcomes.