Write the expansion of π

  π=3.141592653589793238462643

Write the expression for probability

    P(j)=N(j)N        (I)

Here, P(j) is the probability of getting the digit j, N(j) is the number of times the digit j occurs and N is the total number of digits.

Conclusion:

Substitute 0 for N(0) and 25 for N to find P(0)

  P(0)=025=0

Substitute 2 for N(1) and 25 for N to find P(1)

  P(1)=225

Substitute 3 for N(2) and 25 for N to find P(2)

  P(2)=325

Substitute 5 for N(3) and 25 for N to find P(3)

  P(3)=525

Substitute 3 for N(4) and 25 for N to find P(4)

  P(4)=325

Substitute 3 for N(5) and 25 for N to find P(5)

  P(5)=325

Substitute 3 for N(6) and 25 for N to find P(6)

  P(3)=325

Substitute 1 for N(7) and 25 for N to find P(7)

  P(7)=125

Substitute 2 for N(8) and 25 for N to find P(8)

  P(9)=225

Substitute 3 for N(9) and 25 for N to find P(9)

  P(9)=325

Therefore, the probability of getting each of the 10 digits are 0,225,325,525,325,325,325125,225,325 respectively.

The most probable digit is the digit for which P(j) is maximum. From part (a), P(j) is maximum for j=3.

The median is the value of j such that P(X≥j)=P(X≤j).

Write the expression for the average value.

  〈j〉=∑jN(j)N        (II)

Here, 〈j〉 is the expectation value of j.

Conclusion:

From part (b), write the expression for P(X≤4).

  P(X≥4)=025+225+325+525+325=1325

From part (b), write the expression for P(X≥4).

  P(X≤4)=325+325+325+125+225+325=1525

Expand the summation in equation (II) and substitute the corresponding values.

  〈j〉=[0(0)+1(2)+2(3)+3(5)+4(3)+5(3)+6(3)+7(1)+8(2)+9(3)]25=4.72

Therefore, the most probable digit is 3, the median digit is 4 and the average value is 4.72.

Write the expression for the expectation value of j2

  〈j2〉=∑j2N(j)N        (III)

Here, 〈j2〉 is the expectation value of j.

Write the expression for standard deviation

  σ=〈j2〉−〈j〉2        (IV)

Here, σ is the standard deviation.

Conclusion:

Expand the summation in equation (II) and substitute the corresponding values

  〈j2〉=[02(0)+12(2)+22(3)+32(5)+42(3)+52(3)+62(3)+72(1)+82(2)+92(3)]25=28.4

Substitute 28.4 for 〈j2〉 and 4.72 for 〈j〉 in equation (III) to find σ.

  σ=(28.4)−(4.72)2=2.474

Therefore, the standard deviation of the given distribution is 2.474.