The image shows a probability density function (PDF) for a random variable $X$ over the interval $[0, 2]$. The PDF appears to be a uniform distribution, as the height of the function is constant between $0$ and $2$, with a height of $0.5$. This suggests a piecewise uniform distribution.

We need to compute the probability that $X$ is at least 0.2, i.e., $P(X \geq 0.2)$.

Step-by-Step Solution:

1. Recognize the total probability: The total probability for the PDF must integrate to 1. Since the function is constant between 0 and 2, the height of the PDF is $\frac{1}{2}$ (because the area under the curve must be 1, and the length of the interval is 2).

2. Set up the integral: The probability that $X \geq 0.2$ is the area under the curve from $0.2$ to $2$, which can be found by integrating the PDF: $P(X \geq 0.2) = \int_{0.2}^{2} 0.5 \, dx$

3. Evaluate the integral: $P(X \geq 0.2) = 0.5 \times (2 - 0.2) = 0.5 \times 1.8 = 0.9$

Thus, the probability that $X \geq 0.2$ is 0.9.

Let me know if you need further clarification or details!

Here are some follow-up questions to extend this understanding:

1. What is the probability that $X$ is less than or equal to 1?
2. What is the probability that $X$ lies between 0.5 and 1.5?
3. How would the PDF change if the total range was from 0 to 3 instead of 0 to 2?
4. What if the height of the PDF was different, say 0.25? How would the probabilities change?
5. How do you compute the expected value of $X$ for this uniform distribution?

Tip:

For uniform distributions, probabilities are directly proportional to the lengths of intervals within the range, making integrals simple to compute!