Question regarding Monoalphabetic Phi Test 1 &#x2265;<!-- ≥ --> &#x03D5;<!-- ϕ --> (

revistasbibxjm87

revistasbibxjm87

Answered question

2022-05-13

Question regarding Monoalphabetic Phi Test
1 ϕ ( T ) n k k ( n 1 )
Where ϕ ( T ) = i = 1 k n i ( n i 1 ) n ( n 1 ) , T = some text, n= length of text T, k = size of the alphabet, and n i = the frequency of a letter from the alphabet within the text.
Finding the equalities was relatively easy; they exist when the size of the alphabet, k, is equal to one.Where I'm having trouble is proving the inequalities. I can pretty much see that, due to ϕ ( T ) being a sum of frequencies, it'll always be less than one, but should I be proving this with induction? And further, I can't really see how to prove that ϕ ( T ) > n k k ( n 1 ) . Again, I can see how it works, and I've done it with a few examples, but I'm not too sure how to translate that into a rigorous proof.

Answer & Explanation

rynosluv101swv2s

rynosluv101swv2s

Beginner2022-05-14Added 19 answers

This follows immediately from the convexity of x ( x 1 ).
The minimum is achieved when all variables are equal, ie n i = n k . The maximum is achieved at the end points, aka one value equals n and the rest are 0.

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