There is a discrete distribution that is useful in digit counts. It is called th

Question

There is a discrete distribution that is useful in digit counts. It is called the Benford Distribution. Under certain assumptions (which are not important here), the first digit of a number is distributed according to the following discrete distribution: That is, the first digit of a number is a '1' approximately 30.1% of the time. What is the mean digit? What is the median digit? What is the standard deviation of the digits? What is the probability (assuming the above distribution is correct) that the first digit of a random number is even?

Explanation / Answer

Consider:

a)

Thus,  
  
E(x) = Expected value = mean = Sum(xP(x)) =    3.441 [ANSWER]

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b)

For the median, we need to add the probabilities until we exceed 0.500:

As

cdf(2) = 0.301+0.176 = 0.477
cdf(3) = 0.301+0.176+0.125 = 0.602

Then the median = 3. [ANSWER]

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c)

Var(x) = E(x^2) - E(x)^2 =    6.060519

Hence,

s(x) = sqrt [Var(x)] =    2.461812137 [ANSWER]

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d)

P(even) = P(2) + P(4) + P(6) + P(8)

= 0.176+0.097+0.067+0.051

= 0.391 [ANSWER]

x P(x) x P(x) x^2 P(x) 1 0.301 0.301 0.301 2 0.176 0.352 0.704 3 0.125 0.375 1.125 4 0.097 0.388 1.552 5 0.079 0.395 1.975 6 0.067 0.402 2.412 7 0.058 0.406 2.842 8 0.051 0.408 3.264 9 0.046 0.414 3.726

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