A First Course in Probability (10th Edition)
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The equation for a general normal curve with mean ? and standard deviation ? is images
Calculate values for x = 120, 130, ... , 170, 180 where ? = 150 and ? = 30. Note that setting
? = 0 and ? = 1 in this equation gives the equation for the standard normal curve. (Round your answers to four decimal places.)
![### Gaussian (Normal) Distribution Function
The Gaussian distribution, also known as the normal distribution, is a fundamental concept in statistics and probability theory. It is represented by the function:
\[ y = \frac{e^{-\frac{(x - \mu)^2}{2\sigma^2}}}{\sigma\sqrt{2\pi}} \]
#### Explanation of Components:
- \( x \): The variable for which the function is evaluated.
- \( \mu \) (mu): The mean or expectation of the distribution (the center of the peak).
- \( \sigma \) (sigma): The standard deviation, which determines the width of the distribution.
- \( e \): The base of the natural logarithm, approximately equal to 2.71828.
- \( \pi \): Pi, a mathematical constant approximately equal to 3.14159.
#### Understanding the Function:
- The term \( (x - \mu)^2 \) measures the squared deviation of \( x \) from the mean \( \mu \).
- Dividing \( (x - \mu)^2 \) by \( 2\sigma^2 \) scales this deviation relative to the variance, providing a normalized measure of how far \( x \) is from the mean.
- The factor \( \frac{1}{\sigma\sqrt{2\pi}} \) ensures that the total area under the curve sums to 1, which is a requirement for probability distributions.
The Gaussian distribution is symmetric about the mean \( \mu \) and its shape is called a "bell curve." It describes how data points are distributed around the mean—most values cluster around the mean and the probabilities decrease as you move away from the mean in both directions.
This function is widely used in various fields including statistics, natural sciences, and social sciences to represent uncertain or noisy measurements, errors, and other naturally occurring distributions.](https://content.bartleby.com/qna-images/question/373e3400-461c-4f07-afaf-9f92fd60170d/a54d80ed-a435-4c64-8c08-0ee927e3fa89/dhr7zfd_processed.png)
Transcribed Image Text:### Gaussian (Normal) Distribution Function
The Gaussian distribution, also known as the normal distribution, is a fundamental concept in statistics and probability theory. It is represented by the function:
\[ y = \frac{e^{-\frac{(x - \mu)^2}{2\sigma^2}}}{\sigma\sqrt{2\pi}} \]
#### Explanation of Components:
- \( x \): The variable for which the function is evaluated.
- \( \mu \) (mu): The mean or expectation of the distribution (the center of the peak).
- \( \sigma \) (sigma): The standard deviation, which determines the width of the distribution.
- \( e \): The base of the natural logarithm, approximately equal to 2.71828.
- \( \pi \): Pi, a mathematical constant approximately equal to 3.14159.
#### Understanding the Function:
- The term \( (x - \mu)^2 \) measures the squared deviation of \( x \) from the mean \( \mu \).
- Dividing \( (x - \mu)^2 \) by \( 2\sigma^2 \) scales this deviation relative to the variance, providing a normalized measure of how far \( x \) is from the mean.
- The factor \( \frac{1}{\sigma\sqrt{2\pi}} \) ensures that the total area under the curve sums to 1, which is a requirement for probability distributions.
The Gaussian distribution is symmetric about the mean \( \mu \) and its shape is called a "bell curve." It describes how data points are distributed around the mean—most values cluster around the mean and the probabilities decrease as you move away from the mean in both directions.
This function is widely used in various fields including statistics, natural sciences, and social sciences to represent uncertain or noisy measurements, errors, and other naturally occurring distributions.
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