https://file.aiquickdraw.com/mathbot/file/1726792828543vnpuxjnm.jpg

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.

Use the CDF F_Y(y) to find the following probabilities: (a) P[Y < 1], (b) P[Y ≤ 1], (c) P[Y > 2], (d) P[Y ≥ 2], (e) P[Y = 1], (f) P[Y = 3].

This page provides step-by-step solutions for finding probabilities from the CDF of a discrete random variable. Topics include cumulative distribution functions (CDF), probability calculations for discrete values, and the complement rule.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation