x is a gaussian random variable with a PDF as described above, where μ is the mean, σ is the standard deviation , and Fx(X) refers to the cumulative distribution function CDF. It is known that Fx(-3) = 0.500 and Fx(3.7)=0.977, what value of Xo do we find the probability Fx(Xo) = P(X<Xo) = 0.159 ? ### Understanding the Normal DistributionThe diagram illustrates the concept of the normal distribution, a fundamental idea in statistics used to describe how data values are spread across a range. The curve is symmetrical, with most values clustering around the mean, shown as \( \bar{x} \).#### Key Features:1. **Bell Curve Shape**: - The graph is a bell-shaped curve, indicating that the distribution is symmetrical. The highest point on the graph represents the mean of the dataset.2. **Standard Deviations (\(\sigma\))**: - The x-axis is marked in standard deviations from the mean. These are labeled as \(-4\sigma\), \(-3\sigma\), \(-2\sigma\), \(-1\sigma\), 0 (mean), \(+1\sigma\), \(+2\sigma\), \(+3\sigma\), \(+4\sigma\).3. **Probability and Values**: - The y-axis represents the probability of values occurring. Higher points on the curve represent higher probabilities.4. **Probability of Cases**: - Segment probabilities in portions of the curve are shown: - Between \(-2.58\sigma\) and \(-1.96\sigma\) or \(+1.96\sigma\) and \(+2.58\sigma\), the probability is approximately 0.0214. - Between \(-1.96\sigma\) and \(-1.0\sigma\) or \(+1.0\sigma\) and \(+1.96\sigma\), the probability is approximately 0.1359. - Between \(-1.0\sigma\) and \(+1.0\sigma\), the probability is approximately 0.3413, accounting for 68.2% of values.5. **Cumulative Percentage**: - Cumulative percentages for each standard deviation are given below the graph: - \(-4\sigma\) to \(+4\sigma\): Ranges from 0.1% to 99.9%6. **Z Scores and T Scores**: - Z Scores and T Scores align with standard deviations and give numerical values (-4 to +4) representing the distance from the mean, measured in standard deviations.7. **Data Coverage**: - \(-1.96\sigma\)

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x is a gaussian random variable with a PDF as described above, where μ is the mean, σ is the standard deviation , and Fx(X) refers to the cumulative distribution function CDF. It is known that Fx(-3) = 0.500 and Fx(3.7)=0.977, what value of Xo do we find the probability Fx(Xo) = P(X

x is a gaussian random variable with a PDF as described above, where μ is the mean, σ is the standard deviation , and Fx(X) refers to the cumulative distribution function CDF. It is known that Fx(-3) = 0.500 and Fx(3.7)=0.977, what value of Xo do we find the probability Fx(Xo) = P(X

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

### Understanding the Normal Distribution

The diagram illustrates the concept of the normal distribution, a fundamental idea in statistics used to describe how data values are spread across a range. The curve is symmetrical, with most values clustering around the mean, shown as \( \bar{x} \).

#### Key Features:

1. **Bell Curve Shape**:
- The graph is a bell-shaped curve, indicating that the distribution is symmetrical. The highest point on the graph represents the mean of the dataset.

2. **Standard Deviations (\(\sigma\))**:
- The x-axis is marked in standard deviations from the mean. These are labeled as \(-4\sigma\), \(-3\sigma\), \(-2\sigma\), \(-1\sigma\), 0 (mean), \(+1\sigma\), \(+2\sigma\), \(+3\sigma\), \(+4\sigma\).

3. **Probability and Values**:
- The y-axis represents the probability of values occurring. Higher points on the curve represent higher probabilities.

4. **Probability of Cases**:
- Segment probabilities in portions of the curve are shown:
- Between \(-2.58\sigma\) and \(-1.96\sigma\) or \(+1.96\sigma\) and \(+2.58\sigma\), the probability is approximately 0.0214.
- Between \(-1.96\sigma\) and \(-1.0\sigma\) or \(+1.0\sigma\) and \(+1.96\sigma\), the probability is approximately 0.1359.
- Between \(-1.0\sigma\) and \(+1.0\sigma\), the probability is approximately 0.3413, accounting for 68.2% of values.

5. **Cumulative Percentage**:
- Cumulative percentages for each standard deviation are given below the graph:
- \(-4\sigma\) to \(+4\sigma\): Ranges from 0.1% to 99.9%

6. **Z Scores and T Scores**:
- Z Scores and T Scores align with standard deviations and give numerical values (-4 to +4) representing the distance from the mean, measured in standard deviations.

7. **Data Coverage**:
- \(-1.96\sigma\)

Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.

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