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MATH26367 Statistical Methods

Question:
Statistical Methods
Assignment
You can see that Answer exactly 1 of the following 2 questions below, showing all relevant work and the methods used to obtain your results. Answers may only be submitted as a single PDF (.pdf) file. • Submit your answers on Slate by the due date specified on Slate. • This is an individual assignment. Assignments copied in whole or in part will receive a grade of Zero.
Question1:
The same plaintext was encrypted using a monoalphabetic affine cipher, and a polyalphabetic Vigenere cipher. The two resulting ciphertexts (A and B) are:
CiphertextA:

“fvzhpwsggfcpzxwzmbtthjnwdhyxzxbwdszhusnmdbzcnfuuwmfjaeewfvvqsyyhuzcioimlafptuhmqainpriohubpdbvvwpoesiifaqjvloenwhsinvyqszhkdiifaqjvnvyn swskwlvyvbwcafsokfopxuaifpsiahrxszrzhosqqaiyddhywbhytyevtuhydsiagqgitti gtqfrasmggrtvgpraaehytavoltbfiomhyycit”

CiphertextB:

“oyvrvrzdlumtronytgnhteohuoyvroyhuhvrgdolugvgbqtnpzdlotphoyhqmlhavmmoyhroduzhgkrzdlftphlavgzdluqhktgkqhmvhihfytohihuzdlftgoodqhmvhihzdlotph oyhuhkavmmzdlrotzvgfdgkhumtgktgkvrydfzdlydfkhhaoyhutqqvoydmhbdhruhjh jqhutmmvjdeehuvgbvroyhouloygdoyvgbjduh”

a) Use a python Jupyter notebook to compute the IC of each ciphertext (A and B) and determine which one was encrypted using an affine cipher.

b) Use a python Jupyter notebook to construct a histogram of the ciphertext that was encrypted using an affine cipher.

c) Find the most likely private key used for the affine cipher by setting up and solving a system of linear congruences.

d) Use a python Jupyter notebook to decrypt the ciphertext that was encrypted using an affine cipher using the most likely private key.

e) Determine the keyword that was used in the Vigenere cipher.

Question2:

Let x be a... a) Let x be a discrete random variable such that:

P(x) = C(n, x) p x (1 − p) n−x for all 0 ≤ x ≤ n, x ∈ Z.

Show that when p is small, and n is really, Really large, that

P(x) ≈ λ x x! e λ ,

where λ = np.

Hint: when n is really, Really large,

e x ≈ (1 + x n ) n . b)

Let x be a discrete random variable such that:

P(x) = λ x x! e λ for all x ≥ 0, x ∈ Z. Use the fact that e λ = P∞ x=0 λ x x!

To show that the expected value of x is equal to λ.

c) Let x be a discrete random variable with expected value µ.

Show that (x − µ) 2 = x(x − 1) + x − 2xµ + µ 2 .

d) Let x be a discrete random variable such that:

P(x) = λ x x! e λ for all x ≥ 0, x ∈ Z.

Use the fact that e λ = P∞ x=0 λ x x! and that (x − µ) 2 = x(x − 1) + x − 2xµ + µ 2 to show that the variance of x is equal to the expected value of x.