### Understanding Probability Density Functions#### Probability Density Function (PDF)In probability theory, a probability density function (PDF) describes the likelihood of a random variable to take on a particular value. #### Graph Explanation The first graph above shows a function $ y = f(x) $, which is a continuous curve often used to represent a probability density function.#### Question Set**a. If the graph of $ f(x) $ above is a probability density function, the area under $ f(x) $ = 1 (true or false)**- ○ True- ○ False**Explanation**: For a function $ f(x) $ to be a probability density function, the total area under the curve (from $-\infty$ to $+\infty$) must equal 1.The second graph also shows a function $ y = f(x) $ but highlights the area under the curve between two points $ a $ and $ b $.#### Questions and Dropdowns**b. The area under $ f(x) $ on the interval $[a, b]$ is equal to...?**- [Dropdown Selection] - Select an answer - The probability the outcome is greater than $a$ - The probability the outcome is between $a$ and $b$ **Explanation**: In the context of a probability density function, the area under the curve between two points $ a $ and $ b $ represents the probability that the random variable falls within that interval: $ P(a < X < b) $.**c. One method to find the area under $ f(x) $ above is...**- [Dropdown Selection] - Select an answer - Good guesswork - Derivatives - Integration **Explanation**: To find the area under the curve of $ f(x) $ on a defined interval (from $ a $ to $ b $), the method used is integration. This is because the integral of the function $ f(x) $ over the interval $[a, b]$ provides the total area under the curve within those bounds.

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graph of f(x) above is a probabilty density function, the area under f(x) = 1 (true or false) rue alse

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Understanding Probability Density Functions

#### Probability Density Function (PDF)

In probability theory, a probability density function (PDF) describes the likelihood of a random variable to take on a particular value.

#### Graph Explanation

The first graph above shows a function $ y = f(x) $, which is a continuous curve often used to represent a probability density function.

#### Question Set

**a. If the graph of $ f(x) $ above is a probability density function, the area under $ f(x) $ = 1 (true or false)**

- ○ True
- ○ False

**Explanation**: For a function $ f(x) $ to be a probability density function, the total area under the curve (from $-\infty$ to $+\infty$) must equal 1.

The second graph also shows a function $ y = f(x) $ but highlights the area under the curve between two points $ a $ and $ b $.

#### Questions and Dropdowns

**b. The area under $ f(x) $ on the interval $[a, b]$ is equal to...?**

- [Dropdown Selection]
- Select an answer
- The probability the outcome is greater than $a$
- The probability the outcome is between $a$ and $b$

**Explanation**: In the context of a probability density function, the area under the curve between two points $ a $ and $ b $ represents the probability that the random variable falls within that interval: $ P(a < X < b) $.

**c. One method to find the area under $ f(x) $ above is...**

- [Dropdown Selection]
- Select an answer
- Good guesswork
- Derivatives
- Integration

**Explanation**: To find the area under the curve of $ f(x) $ on a defined interval (from $ a $ to $ b $), the method used is integration. This is because the integral of the function $ f(x) $ over the interval $[a, b]$ provides the total area under the curve within those bounds.

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CHECKPOINT 1.4 Sketch left-endpoint and right-endpoint approximations for f(x) = ½ on [1, 2]; use n = 4. Approximate the area using both methods.