To show that the expected value represented by the counter after n INCREMENT operations that have been performed is exactly n.

Given Information: The INCREMENT operation works on a counter containing the value i in a probabilistic manner. If i=2b−1 then it shows an overflow error. Otherwise, it increases the counter by 1 with probability 1/(ni+1−ni) and leaves the counter with probability 1−1/(ni+1−ni) .

Explanation:

The counter after n INCREMENT operations is performed exactly n times for the expected value.

Consider that the initial value of the counter is i and increasing the number representation from ni to ni+1 with a probability of 1ni+1−ni and leaving the value unchanged.

The expected increase is calculated as,

  ni+1−nini+1−ni=1

Hence, the expected increment represented by the counter is 1.

To calculate the variance in the value represented by the register after n INCREMENT operations have been performed.

Given Information: Consider a simple case: ni=100i for all i≥0 and variance of the counter depends on the ni .

Explanation:

Consider that Vn and Xj are the variant and pairwise independent events. The variance Vn is calculated as,

  Var[Vn]=Var[X1]+Var[X2]+⋯Var[Xn]

Since ni=100i and ni+1−ni=100(i+1)−100i=100 . Therefore, the value represented by the counter with probability 99/100 due to the jth INCREMENT is 0 and with the probability of 1/100 .

The value represented increases by 100. Therefore, by the equation (C.27)

  Var[Xj]=E[Xj2]−E2[Xj]=(( 0 2 ⋅ 99 100 )+( 100 2 ⋅ 1 100 ))−12=100−1=99

Now, adding the variances of the Xj gives Var[Vn]=99n .