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Let X be a random variable defined as the sum of N independent Bernoulli trials in which the probability of each Bernoulli taking the value 1 is given by p. The number of Bernoulli trials N is itself a random variable that behaves according to a Poisson distribution function with the parameter.
(a) Derive the conditional distribution function of X given N = n and state your reasoning behind your derivation.
(b) Derive the joint distribution function of X and N and state your reasoning behind your derivation.
(c) Without explicitly calculating it, what would you expect the correlation coecient between X and N to be? (i.e. negative? zero? positive?) Mark sure to provide your reasoning.
(d) Without explicitly calculating the marginal distribution function of X, briefly describe the e↵ect of the Poisson parameter on the mean of X (i.e. what happens to the mean of X as we change ?). Make sure to provide your reasoning.
BONUS: Explicitly derive the marginal distribution function of X
In multiple regression analysis, explain why the typical hypothesis that analysts want to test is whether a particular regression coefficient (B) is equal to zero (H0: B = 0) versus whether that coefficient is not equal to zero (H1: B ≠ 0).
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Use the Crank-Nicolson procedure to solve the following partial differential equation
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There are 2500 labor hours and 4,000 square feet of lumber available each week. Determine the product mix that maximizes weekly profit.
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