Question

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.

a.The expected value is sum of all possible leading digits divided by the number of possible leading digits.

b. The expected value is the average value of the leading significant digits in any set of numerical values that follow Benford's law.

c. The expected value is the most common value of the leading significant digit in a set of numerical values.

d. The expected value is sum of the probabilities associated with each leading digit divided by the number of possible leading digits.

e.The expected value is the long‑run average value of the leading significant digits of a set of numerical values.

Answer #1

**Solution:**

We are given that: probability distribution of the leading significant digit of a randomly selected data value in a dataset that behaves according to Benford's law.

x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

P(x) | 0.301 | 0.176 | 0.125 | 0.097 | 0.079 | 0.067 | 0.058 | 0.051 | 0.046 |

We have to find Expected value of x. that is E(X)=........?

To find expected value, we need to find following table:

x | P(x) | x * P(x) |
---|---|---|

1 | 0.301 | 0.301 |

2 | 0.176 | 0.352 |

3 | 0.125 | 0.375 |

4 | 0.097 | 0.388 |

5 | 0.079 | 0.395 |

6 | 0.067 | 0.402 |

7 | 0.058 | 0.406 |

8 | 0.051 | 0.408 |

9 | 0.046 | 0.414 |

Thus Expected value is:

The statement that best describes the interpretation of the expected value of the Benford's law probability distribution is:

**e.The expected value is the long‑run average value of
the leading significant digits of a set of numerical
values.**

The first significant digit in any number must be 1, 2, 3, 4,
5, 6, 7, 8, or 9. It was discovered that first digits do not occur
with equal frequency. Probabilities of occurrence to the first
digit in a number are shown in the accompanying table. The
probability distribution is now known as Benford's Law. For
example, the following distribution represents the first digits in
219 allegedly fraudulent checks written to a bogus company by an
employee attempting to...

The first significant digit in any number must be 1, 2, 3,
4,5, 6, 7, 8, or 9. It was discovered that first digits do not
occur with equal frequency. Probabilities of occurrence to the
first digit in a number are shown in the accompanying table. The
probability distribution is now known as Benford's Law.
Forexample, the following distribution represents the first digits
in 232 allegedly fraudulent checks written to a bogus company by an
employee attempting to embezzle funds...

Benford's Law states that the first nonzero digits of numbers
drawn at random from a large complex data file have the following
probability distribution.† First Nonzero Digit 1 2 3 4 5 6 7 8 9
Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
Suppose that n = 275 numerical entries were drawn at random from a
large accounting file of a major corporation. The first nonzero
digits were recorded for the sample. First Nonzero Digit 1 2...

In a trial, the defendant was accused of issuing checks to a
nonexistent vendor. The amounts of the checks are listed in the
accompanying data table in order by row. When testing for
goodness-of-fit with the proportions expected with Benford's law,
it is necessary to combine categories because not all expected
values are at least 5. Use one category with leading digits of 1,
a second category with leading digits of 2, 3, 4, 5, and a third
category with...

Recall that Benford's Law claims that numbers chosen from very
large data files tend to have "1" as the first nonzero digit
disproportionately often. In fact, research has shown that if you
randomly draw a number from a very large data file, the probability
of getting a number with "1" as the leading digit is about 0.301.
Now suppose you are an auditor for a very large corporation. The
revenue report involves millions of numbers in a large computer
file....

Recall that Benford's Law claims that numbers chosen from very
large data files tend to have "1" as the first nonzero digit
disproportionately often. In fact, research has shown that if you
randomly draw a number from a very large data file, the probability
of getting a number with "1" as the leading digit is about 0.301.
Now suppose you are an auditor for a very large corporation. The
revenue report involves millions of numbers in a large computer
file....

Recall that Benford's Law claims that numbers chosen from very
large data files tend to have "1" as the first nonzero digit
disproportionately often. In fact, research has shown that if you
randomly draw a number from a very large data file, the probability
of getting a number with "1" as the leading digit is about 0.301.
Now suppose you are an auditor for a very large corporation. The
revenue report involves millions of numbers in a large computer
file....

Construct a probability distribution for the
following data set, which represents the number of hours of
students in a college slept previous night. Let the random variable
x represent the number of hours of sleep.
Hours
4
5
6
7
8
9
10
Students
1
6
11
18
9
3
2
b) Calculate the expected value (mean), variance, and
standard deviation of the probability distribution.

Recall that Benford's Law claims that numbers chosen from very
large data files tend to have "1" as the first nonzero digit
disproportionately often. In fact, research has shown that if you
randomly draw a number from a very large data file, the probability
of getting a number with "1" as the leading digit is about 0.301.
Now suppose you are the auditor for a very large corporation. The
revenue file contains millions of numbers in a large computer data...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 3 minutes ago

asked 5 minutes ago

asked 6 minutes ago

asked 9 minutes ago

asked 11 minutes ago

asked 11 minutes ago

asked 27 minutes ago

asked 37 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 1 hour ago