Consider the first 25 digits in the decimal expansion of π (3, 1, 4, 1, 5, 9, … ).
(a) If you selected one number at random, from this set, what are the probabilities of getting each of the 10 digits?
(b) What is the most probable digit? What is the median digit? What is the average value?
(c) Find the standard deviation for this distribution.
From Math Tables: π = 3:141592653589793238462643……
(a) P(0) = 0 P(1) = 2=25 P(2) = 3=25 P(3) = 5=25 P(4) = 3=25
P(5) = 3=25 P(6) = 3=25 P(7) = 1=25 P(8) = 2=25 P(9) = 3=25
In general, P(j) = \frac{N(j)}{N} .
(b) Most probable: 3. Median: 13 are ≤ 4, 12 are ≥ 5, so median is 4.
Average: \left\langle j\right\rangle = \frac{1}{25}[0 . 0 + 1 . 2 + 2 . 3 + 3 . 5 + 4 . 3 + 5 .3 + 6 . 3 + 7 . 1 + 8 . 2 + 9 . 3] .
= \frac{1}{25} [0 + 2 + 6 + 15 + 12 + 15 + 18 + 7 + 16 + 27] = \frac{118}{25} = 4.72 .
(c) \left\langle j^2\right\rangle = \frac{1}{25} [0 + 1^2 . 2 + 2^2 . 3 + 3^2 . 5 + 4^2 . 3 + 5^2 . 3 + 6^2 . 3 + 7^2 . 1 + 8^2 . 2 + 9^2 . 3] .
= \frac{1}{25} [0 + 2 + 12 + 45 + 48 + 75 + 108 + 49 + 128 + 243] = \frac{710}{25} = 28.4 .
σ^2 = \left\langle j^2\right\rangle – \left\langle j\right\rangle^2 = 28.4 -4.72^2 = 28.4 – 22.2784 = 6.1216 , σ = \sqrt{6.1216} = 2.474 .