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Assume that X = (X1, …., Xn) is a vector of independently observations of the Bernoulli distribution Xi ∼ Be(θ) where the parameter θ ∈ [0, 1] is unknown.

1. Decide the logarithim probability function L(θ; x) = log fn(x|θ) if n = 9, and x = (1, 0, 0, 1, 1, 0, 1, 0, 0).

2. Maximize L(θ; x) = log fn(x|θ) by taking θ into account.

(Standalone) 3. Assume that X1, X2, and X3 are randomly and evenly sampled from a distribution, where the median is equal to the unknown parameter θ. Calculate the probability P(min{X1, X2, X3} < θ < max{X1, X2, X3}). Formulate your response as a statement of the confidence interval for θ.

For n ≥ 1, assume that a lazy person would randomly insert n differently addressed letters into n envelopes without checking that the addresses on the envelopes and letters coincide. Let Xn be the amount of letters that reach the right recipient.

4. Decide the expected value and variance in the variable Xn. (Hint: you can write Xn as a sum of dependent Bernoulli variables, where the covariance can be decided using combinatorics.)

5. Decide towards which distribution Xn is converging towards when n approaches infinity.

Assume that X is normally disctibuted with the expected value 0 and the standard deviation 1. Assume that Y is a variable so that the generated function is given by: E (e^(sY) | X) = e^(sX^(2)+3s)

6. What is the expected value of Y?

7. What is the variance of Y and the covariance Cov(X, Y)?

p(7)

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