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(-4.8) = 0.500 and Fx(1.4)=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(-4.8) = 0.500 and Fx(1.4)=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(-4.8) = 0.500 and Fx(1.4)=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(-4.8) = 0.500 and Fx(1.4)=0.977, what value of Xo do we find the probability Fx(Xo) = P(X<Xo) = 0.159 ? ( Ans = -7.9)
Transcribed Image Text:The Normal Distribution is represented as a symmetrical, bell-shaped curve centered around the mean (\(\overline{X}\)). The x-axis denotes values, and the y-axis shows probability.
### Key Features:
- **Center**: The curve is highest at the mean (\(\overline{X}\)).
- **Symmetry**: The distribution is symmetric about the mean.
- **Tails**: The tails of the distribution extend infinitely, approaching but never touching the x-axis.
### Probability and Standard Deviations:
- **-1.96σ to +1.96σ**: Approximately 95% of the values fall within this range.
- **-2.58σ to +2.58σ**: Around 99% of the values are within this range.
### Probability of Cases:
- **Outside ±3σ**: Probability is approximately 0.0013.
- **Between ±2σ to ±3σ**: Probability is around 0.0214.
- **Between ±1σ to ±2σ**: Probability is approximately 0.1359.
- **Between 0σ and ±1σ**: Probability is about 0.3413.
### Cumulative Percentages:
- **±1σ**: Accounts for 68.3% of the data.
- **±2σ**: Accounts for 95.5% of the data.
- **±3σ**: Covers about 99.7% of the data.
### Z Scores and T Scores:
The x-axis also shows corresponding Z scores and T scores for each standard deviation, ranging from -4.0 to +4.0.
This graph is essential for understanding data distributions, calculating probabilities, and performing statistical analyses in various fields.
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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