The image provides a cumulative distribution function (CDF) $F_Y(y)$ for a random variable $Y$, and you are tasked with calculating certain probabilities based on the given CDF.

Steps to Solve:

The CDF $F_Y(y)$ provides the cumulative probability up to a certain value of $y$. The probability of any specific event can be derived by examining the differences between CDF values.

1. Reading the CDF $F_Y(y)$ graph:

• At $y = 0$, $F_Y(0) = 0$
• At $y = 1$, $F_Y(1) = 0.2$
• At $y = 2$, $F_Y(2) = 0.5$
• At $y = 3$, $F_Y(3) = 0.7$
• At $y = 4$, $F_Y(4) = 0.9$
• At $y = 5$, $F_Y(5) = 1.0$

Now, let's answer the given questions:

(a) $\mathbb{P}(Y < 1)$

This is the probability that $Y$ is less than 1. Using the CDF: $\mathbb{P}(Y < 1) = F_Y(1) = 0.2$

(b) $\mathbb{P}(Y \leq 1)$

This is the probability that $Y \leq 1$. From the CDF: $\mathbb{P}(Y \leq 1) = F_Y(1) = 0.2$

(c) $\mathbb{P}(Y > 2)$

This is the probability that $Y$ is greater than 2. Using the complementary rule: $\mathbb{P}(Y > 2) = 1 - F_Y(2) = 1 - 0.5 = 0.5$

(d) $\mathbb{P}(Y \geq 2)$

For $Y \geq 2$, the probability is: $\mathbb{P}(Y \geq 2) = 1 - F_Y(1) = 1 - 0.2 = 0.8$

(e) $\mathbb{P}(Y = 1)$

The probability of $Y = 1$ is the difference between $F_Y(1)$ and $F_Y(0)$: $\mathbb{P}(Y = 1) = F_Y(1) - F_Y(0) = 0.2 - 0 = 0.2$

(f) $\mathbb{P}(Y = 3)$

The probability of $Y = 3$ is the difference between $F_Y(3)$ and $F_Y(2)$: $\mathbb{P}(Y = 3) = F_Y(3) - F_Y(2) = 0.7 - 0.5 = 0.2$

Summary of Probabilities:

• (a) $\mathbb{P}(Y < 1) = 0.2$
• (b) $\mathbb{P}(Y \leq 1) = 0.2$
• (c) $\mathbb{P}(Y > 2) = 0.5$
• (d) $\mathbb{P}(Y \geq 2) = 0.8$
• (e) $\mathbb{P}(Y = 1) = 0.2$
• (f) $\mathbb{P}(Y = 3) = 0.2$

Would you like more details or have any questions on these calculations?

Follow-up Questions:

1. What is the difference between $\mathbb{P}(Y < y)$ and $\mathbb{P}(Y \leq y)$?
2. How can you compute probabilities for a continuous random variable from its CDF?
3. Can the CDF ever decrease? Why or why not?
4. What is the relationship between a CDF and the probability density function (PDF)?
5. How does one compute the probability of an interval, like $\mathbb{P}(2 < Y \leq 4)$, from the CDF?

Tip: Remember, the CDF $F_Y(y)$ gives the probability that $Y$ is less than or equal to $y$. For exact values, use differences in CDF values at specific points.