Xi'an's Og

The weekend riddle from The Riddler is rather straightforward [my wording and simplification]:

Construct a decimal number X between 0 and 1 by drawing the first digit a¹ uniformly over {0,1,…,9}, the second digit a² uniformly over {0,1,…,9}, &tc., until 0 is attained. What is the expectation of this random variable X?

Since each new digit has expectation half of the previous digit, the expectation is an infinite geometric series with common ratio 20⁻¹ and factor 9, leading to an expectation of 9/19. As verified with the following R code:

skr<-function(){
  a=9;b=0
  while((a<-sample(rep(0:a,2),1))>0)b=10*b+a
  while(b>=1)b=b/10
  return(b)}

This entry was posted on December 2, 2019 at 12:19 am and is filed under Books, Kids, R with tags geometric series, riddle, The Riddler, weekend. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.