Answer by lonza leggiera for How to prove the expectation of the number of...
By definition,$$\mathbb{E}(f(X))=\frac{\displaystyle\sum_{j=1}^mf(j)}{m}\ .$$Now let$$a_{ij}=\cases{0 & if the binary representation of $\ j$ \\&has fewer than $\ i\ $ trailing zeroes\\1&...
View ArticleAnswer by r.e.s. for How to prove the expectation of the number of trailing...
There are exactly $2^{b-1}$ positive $b$-bit numbers (i.e. the numbers $2^{b-1},..,2^b-1$) for $b=1,2,3,...$, so let's consider $m=2^b-1$.Let $N(\le b,t)$ be the number of positive numbers with $\le b$...
View ArticleHow to prove the expectation of the number of trailing zeros
Let $X$ be uniformly sampled from the integers $\{1, \dots, m\}$ for $m > 0$. For $x>0$, we define $f(x)$ to be the number of trailing zeros in the binary representation of $x$.What...
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