課程名稱：微積分 開課教師：陸行教授 開課系級：金融一 考試日期(年月日)：2011/6/1 考試時限(Mins)：60(mins) 試題本文 1.A roulette wheel has 18 red numbers, 18 black numbers, and 2 green numbers. Each time the wheel is spun, George bets that the ball will land on a red number. When the ball lands on red, he wins $1 from the casino for every $1 bet. When the ball lands on a black or green number, he loses his bet. Find his expected winnings from a $1 bet on red. 2.George,the roulette player in question 1,finds another wheel with the same number of red and black numbers but with only 1 green number. Now what are his expected winnings from a $1 bet on red. 3.In section 9.3, it is shown that 1+2x+3x^2+...=1/(1-x)^2 for -1<x<1. Use this fact to show that a geometric random variable X with parameter p has expected value E(X)=1/p. 4.In this question, either find a number k such that the given function is a probability density function or explain why no such number exists. f(x)={x^3+kx for 0<=x<=1 {0 otherwise 5.In this question, f(x) is a probability density function for a particular random variable X. Use integration to find the indicated probabilities. f(x)={0.1*e^-0.1x for x >=0 {0 for x<0 (a). P(0<=X<=+∞) (b). P(X<=2) (c). P(X>=5) 6.In this question, the probability density function for a continuous random variable X is given. Find the expected value E(X) and the variance Var(X) of X. f(x)={3/x^4 for 1<=x<∞ {0 for x<1 7.In this question, the density function f(x) of a normal random variable X is given. In each case, find the expected value E(X), the variance Var(X), and the standard deviation σ(X). f(x)={1/(2π)^0.5}*e^-0.5*(x+6)^2 --

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