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spring98exam2[1]

# Let z denote the total amount of money that alice

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Unformatted text preview: te the total amount of money that Alice collects. Find the expected value and the variance of Z. Answer: The random variable Y includes Alice's last miss. Therefore the number of darts that she gets in the bullseye is Y ; 1, and not Y . Hence Z = 20 + 70(Y ; 1) = 70Y ; 50. From this, we know that Var Z ] = 702 Var Y ] = 4900 E Z ] = 20 + 70 E Y ; 1] = 20 + 70 2 = 40 7 18 = 1800 49 E Z ] = 40 Var Z ] = 1800 Problem 4 (20 points) A continuous random variable X is uniformly distributed on the interval in ;1 +1]. a. Find the probability density function of the random variable Y = X 2 . Answer: The pdf of X is 81 < if ; 1 u 1 fX (u) = : 2 0 otherwise We see that the range of values Y can take is 0 1]. Therefore the CDF of Y , given by FY (v), is equal to 1 for v > 1. For 0 v 1, FY (v) = P fX 2 vg = P f; v X + vg pv) ; F (;pv) = 2pv = pv = F (+ X X p p Therefore the pdf of Y is given as 2 8 1 < p fY (v) = : 2 v 0 0 v 1 otherwise fY (v) = 8 > > 1 > p > <2 v > >0 > > : 0 v 1 otherwise b. Let the random variable S be the sign of X , that is S = 1 if X 0 and S = ;1 if X < 0. Find the mean and the variance of the random variable Z = 1 ; S 2. Answer: The trick here is to observe that Z = 0 with probability 1 | it is a deterministic quantity! Therefore E Z ] = 0 and Var Z ] = 0. E Z] = 0 Var Z ] = 0...
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