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.

First 25 decimal digits of π is,

$\mathrm{\pi }=3.141592653589793238462643...$

Part (a)

The probability of choosing a particular digit is the frequency, divided by the total number of digits

$\begin{array}{cc}0:\frac{0}{25}=0& 5:\frac{3}{25}=0.12\\ 1:\frac{2}{25}=0.08& 6:\frac{3}{25}=0.12\\ 2:\frac{3}{25}=0.12& 7:\frac{1}{25}=0.04\\ 3:\frac{5}{25}=0.20& 8:\frac{2}{25}=0.08\\ 4:\frac{3}{25}=0.12& 9:\frac{3}{25}=0.12\end{array}$

Part (b)

$\{1,1,2,2,2,3,3,3,3,3,4,4,4,5,5,5,6,6,6,7,8,8,9,9,9\}$

The highlighted number in this set represents the median

Median = 4

Most probable digit chosen is 3 since it’s the number with highest frequency

The average value is given by the expectation value.

Thus, the most probable digit, median, and average value are 4, 3, and 28.4 respectively.

Standard deviation is given by

$\begin{array}{l}\mathrm{\sigma }=\sqrt{⟨{\mathrm{j}}^2⟩-⟨\mathrm{j}{⟩}^2}\\ \mathrm{\sigma }=\sqrt{28.4-4{.72}^2}\\ \mathrm{\sigma }=2.47\end{array}$

Thus, the standard deviation is 2.47.