Question

Fraudulent numbers in tax returns, payment records, invoices, etc. often display patterns that aren’t present in legitimate records. It is a striking fact that the first digits of numbers in legitimate records often have probabilities that follow the model (known as Benford’s Law) partially shown in the following probability distribution, where the random variable x is the first digit of the number. x 1 2 3 4 5 6 7 8 9 P(x) 0.301 0.176 0.125 ? 0.079 0.067 0.058 0.051 0.046 (a) Find the probability that the first digit is a 4. (b) Let A be the event that the first digit is a 1 or a 2; B be the event that the first digit is more than 6 and C be the event that the first digit is an odd number. Find P(A), P(B) and P(C). (c) Find the probabilities for the following events i. P(A and B) ii. P(A or B) iii. P(A and B) iv. P(B|C) (d) Based on your result from part (i.) above, what can you say about the relationship between events A and B? (e) Find the mean and standard deviation for the probability distribution.

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It is a striking fact that the first digits of numbers in
legitimate records often follow a distribution known as
Benford's Law, shown below.
First digit
1
2
3
4
5
6
7
8
9
Proportion
0.285
0.186
0.126
0.084
0.069
0.067
0.034
0.039
0.110
Fake records usually have fewer first digits 1, 2, and 3. What
is the approximate probability, if Benford's Law holds, that among
1153 randomly chosen invoices there are no more than 670 in amounts
with...

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...

It is a striking fact that the first digits of numbers in
legitimate records often follow a distribution known as
Benford's Law, shown below.
First digit
1
2
3
4
5
6
7
8
9
Proportion
0.28
0.152
0.129
0.088
0.05
0.07
0.042
0.035
0.154
Fake records usually have fewer first digits 1, 2, and 3. What
is the approximate probability, if Benford's Law holds, that among
1189 randomly chosen invoices there are no more than 688 in amounts
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 the auditor for a very large corporation. The
revenue file contains millions of numbers in a large computer data...

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...

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....

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