II. Essay (30%)
"Eye contact is a social signal." Do you agree or disagree with this statement? Write an essay of about 300 words to explain your reasons. Provide at least three examples to support your claim.

(a) Find the mean value and variance of Y₁ = X₁ + X₂.Answer: Mean = __( A )__ ； Variance =__( B )__

(b) Find the mean value and variance of of Y₂ = 2X₁.Answer: Mean = __(C)__ ； Variance =__(D)__

(a) The (marginal) probability mass function of X, pₓ(x) =__(E)__.

(b) The (marginal) probability mass function of Y, pᵧ(y) =__(F)__.

(c) Are X and Y independent? Answer=__(G)__.

(a) The expected value of X, E(X) = __(H)__.

(b) The expected value of Y, E(Y) =__(I)__.

(c) The covariance, Cov(X,Y) =__(J)__.

(a) If two or more of the six devices in the Administration building are activated during the same weekend, the system automatically signals the police. What is the probability that this system will signal the police next weekend?Answer=__(K)__. Hint: remember the assumption.

(b) An intruder has a 5% chance of not activating the device located in the President's office.Campus police has obtained information about a student prank in preparation and, as a result, estimates that there is a 20% chance that an intruder will visit the President's office during next Columbus Day weekend. If this device is activated during that weekend, what is the probability that there is an intruder in the President's office? Answer=__(L)__

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