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Let X be a random variable that denotes the position of the character where COVFEFE occurs for the first time. For example, if COVFEFE are the first seven characters typed then X would be 7. If F is the probability mass function of X, we may write F as:

$F(x) = 0; \quad 0 \le x < 7$, as COVFEFE cannot appear until 7 characters are typed

$F(7) = k, \quad where \: k= 1/26^7$, all characters are chosen randomly from 26 characters

$F(x) = (1- (F(0)+F(1)+ ... + F(x-7)))*k, \quad x\ge7$, COVFEFE shouldn't appear at any position until x-7 and the next 7 letters should match exactly. All events are mutually exclusive so their probability may be added.

This can be re-written as:

$F(x) = F(x-1) - k*F(x-7); \quad x\ge8$, using the above two statements

Now, the question asks for expected value of X.
\[
E[X] = \sum_{x=0}^\infty xF(x) = \sum_{x=7}^\infty xF(x) \\
\implies E[X] = 7F(7) + \sum_{x=8}^\infty xF(x) \\
\implies E[X] = 7k + \sum_{x=8}^\infty x(F(x-1) - kF(x-7)) \\
\implies E[X] = 7k + \sum_{x=8}^\infty xF(x-1) - k\sum_{x=8}^\infty xF(x-7) \\
\implies E[X] = 7k + \sum_{x=8}^\infty (x-1+1)F(x-1) - k\sum_{x=8}^\infty (x-7+7)F(x-7) \\
\implies E[X] = 7k + \sum_{x=8}^\infty (x-1)F(x-1) + \sum_{x=8}^\infty F(x-1) - k\sum_{x=8}^\infty (x-7)F(x-7) - 7k\sum_{x=8}^\infty F(x-7) \\
\implies E[X] = 7k + \sum_{x=7}^\infty xF(x) + \sum_{x=7}^\infty F(x) - k\sum_{x=1}^\infty xF(x) - 7k\sum_{x=1}^\infty F(x) \\
\implies E[X] = 7k + E[X] + \sum_{x=7}^\infty F(x) - kE[X] - 7k\sum_{x=1}^\infty F(x) \\
\implies kE[X] = 7k + \sum_{x=0}^\infty F(x) - 7k\sum_{x=0}^\infty F(x) \\
\implies kE[X] = 7k + (1-7k)\sum_{x=0}^\infty F(x)
\]
Clearly F being a probability mass function, $\Sigma_{x=0}^\infty F(x) = 1 $
\[
\implies kE[X] = 7k + 1 - 7k \\
\implies kE[X] = 1 \\
\implies E[X] = 1/k = 26^7
\]

Sun, 03 Dec 2017 12:58 GMT