Benford's law, also known as the first‑digit law, represents a probability distribution of the leading significant digits of numerical values in a data set. A leading significant digit is the first occurring non‑zero integer in a number. For example, the leading significant digit in the number 127127 is 11. Let this leading significant digit be denoted ?x. Benford's law notes that the frequencies of ?x in many datasets are approximated by the probability distribution shown in the table. ?x 11 22 33 44 55 66 77 88 99 ?(?)P(x) 0.3010.301 0.1760.176 0.1250.125 0.0970.097 0.0790.079 0.0670.067 0.0580.058 0.0510.051 0.0460.046 Determine ?(?)E(X), the expected value of the leading significant digit of a randomly selected data value in a dataset that behaves according to Benford's law? Please give your answer to the nearest three decimal places. ?(?)E(X) = Select the statement that best describes the interpretation of the expected value of the Benford's law probability distribution. The expected value is the leading significant digit that shows up the most in a set of numerical values. The expected value is calculated by taking an unweighted average of all values of ?x. The expected value is the average value of the leading significant digits in any set of numerical values that follow Benford's law. The expected value is sum of the probabilities associated with each leading digit divided by the number of possible leading digits. The expected value is the long‑run average value of the leading significant digits of a set of numerical values.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Benford's law, also known as the first‑digit law, represents a probability distribution of the leading significant digits of numerical values in a data set. A leading significant digit is the first occurring non‑zero integer in a number. For example, the leading significant digit in the number 127127 is 11. Let this leading significant digit be denoted ?x.
Benford's law notes that the frequencies of ?x in many datasets are approximated by the probability distribution shown in the table.
| ?x | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 |
|---|---|---|---|---|---|---|---|---|---|
| ?(?)P(x) | 0.3010.301 | 0.1760.176 | 0.1250.125 | 0.0970.097 | 0.0790.079 | 0.0670.067 | 0.0580.058 | 0.0510.051 | 0.0460.046 |
Determine ?(?)E(X), the
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