Question

i have uploaded the example 3.23 it may help

Policies on cryptocurrency mining typically decide which N to use in order to control the rate at which co ins are generated (see Examplem), and increase the N over time to control the rate of growth. (a) If (even) N is increased by 50%, write an expression for the probability distribution of the number of attempted strings will be tried by miners before the reward is released. (b) Plot the distributions for N = 100 and for N = 150 on a single graph with R. Example 3.23 (Simplified cryptocurrency mining). For many cryptocurrencies, such as BitCoin, once the previous unit of currency has been created, a random string is generated. The first person to guess the string is rewarded with new coins. The people doing the guessing ("miners") run code on their computers that randomly guess the string over and over, where each guess is equally likely to be correct or incorrect (as far as the k in an attempt to get the reward. If the number of possible strings that can be generated is N (typically, a very large number), then the probability that each guess is correct is and the guesses may be assumed to be independent on a small scale, so the number of guesses before the coins are unlocked is a geometric random variable Thus, the probability that it takes k+1 attempts from miners to unlock the reward, we have

Answer 1

(a)

If N is increased by 50% the new number of possible strings that can be generated is equal to (N + 50% of N) = 1.5N

Thus, the probability that it takes k+1 attempts from miners to unlock the reward, or the probability that the number of strings attempted before the reward is unlocked is equal to k is given by:

$P(X=k) = \begin{pmatrix}1 - \frac{1}{1.5N} \end{pmatrix}^k \begin{pmatrix} \frac{1}{1.5N}\end{pmatrix} \ \ \ ; k = 0,1,2,3,...$

(b)

The R code for plotting the distributions is given below:
> k=0:250
> y=(1-1/100)^k*(1/100)
> z=(1-1/150)^k*(1/150)
> plot(k,y,type='p',col="red",ylab="P(X=k)")
> points(k,z,col="blue")
> legend(150,0.010,pch=1,col=c("red","blue"),legend=c("N = 100","N = 150"))

The output plot is given below:

髦3 髦3 N=100 N = 150 髦3 f1 髦3 250 200 150 100 50 0

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