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1. Daily Airlines ies from Amsterdam to London every day. The price of a ticket for this extremely popular ight route is $75. The aircraft has a passenger capacity of 150. The airline management has made a policy to sell 160 tickets for this ight in order to protect themselves against noshow passengers. Experience has shown that the probability of a passenger being a no-show is equal to 0.1. The booked passengers act independently of each other. Given this overbooking strategy, what is the probability that some passengers will have to be bumped from the ight? (You can either use any computing tool to obtain a numerical value, or simply leave an expression simpli_ed to your best.)

2. Find an ex in which three random variables are pairwise independent but not independent as a family.

3. Suppose that X and Y are two dependent random variables. In statistical applications, it is often the case that we can observe the random variable X but we want to know the dependent random variable Y . What is the best linear predictor of Y with respect to X? In other words, for which linear function y = α +βx is

E[(Y -βX)2]

minimal? Express your answer in terms of: μX = E(X), μY = E(Y ), σX = √ Var(X), σY = √Var(Y ), and ρXY = ρ(X; Y ).

4. Let X and Y have joint mass function

f(j, k) = (c(j + k)aj+k /j!k!) , j, k>= 0

where a is a constant. Find c, P(X = j), P(X + Y = r), and E(X).

5. Let X1, X2,..... be identically distributed random variables with mean μ, and let N be a random variable taking values in the non-negative integers and independent of the Xi. Let S = X1 + X2 +....... + XN. Show that

E(S|N) =  μN, and deduce that E(S) =  μE(N).

6. A factory has produced n robots, each of which is faulty with probability Φ. To each robot a test is applied which detects the fault (if present) with probability δ. Let X be the number of faulty robots, and Y the number detected as faulty. Assuming the usual independence, show that

E(XjY ) = (nΦ(1 - δ) + (1 -Φ)Y)/ (1 -Φδ)

 

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