MATH26367 Statistical Methods

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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.

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

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

where λ = np.

Hint: when n is really, Really large,

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:

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.