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(-0.7) = 0.500 and Fx(1.3)=0.841, what value of Xo do we find the probability Fx(Xo) = P(X<Xo) = 0.023 ? ### Understanding the Normal DistributionThe graph illustrates the Normal Distribution, a fundamental concept in statistics representing how values are distributed around the mean (average), denoted here by \( X \).#### Key Components:- **Probability vs. Values**: The vertical axis represents probability, while the horizontal axis represents values.- **Bell Curve Shape**: The curve is symmetric, centered around the mean, and describes how data points are spread. Most of the data is close to the mean.- **Standard Deviations (\( \sigma \))**: - \( -4\sigma \) to \( +4\sigma \) along the horizontal axis represent standard deviations from the mean. - The percentages below the axis indicate cumulative probabilities at each standard deviation. - Cumulative percentages show: - \( -1\sigma \) to \( +1\sigma \): 68.2% of values - \( -2\sigma \) to \( +2\sigma \): 95.4% of values - \( -3\sigma \) to \( +3\sigma \): 99.7% of values- **Z Scores and T Scores**: - Displayed below the standard deviations, these are used in statistical analysis to understand data distribution.#### Probability of Cases:- Represented by shaded regions under the curve, showing probabilities in terms of portions of the curve. - \( \approx 0.3413 \): Percentage of data within ±1 standard deviation - \( \approx 0.1359 \): Between 1-2 standard deviations - \( \approx 0.0214 \): Between 2-3 standard deviations - \( \approx 0.0013 \): Beyond 3 standard deviations ### Practical Implications:Understanding the normal distribution assists in various fields such as psychology, finance, and natural sciences for hypothesis testing, decision making, and predicting patterns. It serves as the foundation for many statistical tests and is essential for data analysis.

For the given Gaussian distribution, the parameters are specified as follows:The mean, μ.The…

Answered: x is a gaussian random variable with a…

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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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(-0.7) = 0.500 and Fx(1.3)=0.841, what value of Xo do we find the probability Fx(Xo) = P(X<Xo) = 0.023 ?

### Understanding the Normal Distribution

The graph illustrates the Normal Distribution, a fundamental concept in statistics representing how values are distributed around the mean (average), denoted here by \( X \).

#### Key Components:

- **Probability vs. Values**: The vertical axis represents probability, while the horizontal axis represents values.

- **Bell Curve Shape**: The curve is symmetric, centered around the mean, and describes how data points are spread. Most of the data is close to the mean.

- **Standard Deviations (\( \sigma \))**:
- \( -4\sigma \) to \( +4\sigma \) along the horizontal axis represent standard deviations from the mean.
- The percentages below the axis indicate cumulative probabilities at each standard deviation.
- Cumulative percentages show:
- \( -1\sigma \) to \( +1\sigma \): 68.2% of values
- \( -2\sigma \) to \( +2\sigma \): 95.4% of values
- \( -3\sigma \) to \( +3\sigma \): 99.7% of values

- **Z Scores and T Scores**:
- Displayed below the standard deviations, these are used in statistical analysis to understand data distribution.

#### Probability of Cases:

- Represented by shaded regions under the curve, showing probabilities in terms of portions of the curve.
- \( \approx 0.3413 \): Percentage of data within ±1 standard deviation
- \( \approx 0.1359 \): Between 1-2 standard deviations
- \( \approx 0.0214 \): Between 2-3 standard deviations
- \( \approx 0.0013 \): Beyond 3 standard deviations

### Practical Implications:

Understanding the normal distribution assists in various fields such as psychology, finance, and natural sciences for hypothesis testing, decision making, and predicting patterns. It serves as the foundation for many statistical tests and is essential for data analysis.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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