An Elementary Introduction to Mathematical Finance, Third by Sheldon M. Ross

By Sheldon M. Ross

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3 Argue (a picture is acceptable) that P{|Z | > x} = 2P{Z > x}, where x > 0 and Z is a standard normal random variable. 4 Let X be a normal random variable having expected value μ and variance σ 2 , and let Y = a+bX. Find values a, b (a = 0) that give Y the same distribution as X. Then, using these values, find Cov(X, Y ). 2. (a) Specify an interval in which the blood pressures of approximately 68% of the adult male population fall. (b) Specify an interval in which the blood pressures of approximately 95% of the adult male population fall.

15 Prove: (a) (b) (c) (d) Cov(X, Y ) = Cov(Y, X ); Cov(X, X ) = Var(X ); Cov(cX, Y ) = c Cov(X, Y ); Cov(c, Y ) = 0. 16 If U and V are independent random variables, both having variance 1, find Cov(X, Y ) when X = aU + bV, Y = cU + dV. 17 If Cov(X i , X j ) = ij, find (a) Cov(X1 + X 2 , X 3 + X 4 ); (b) Cov(X1 + X 2 + X 3 , X 2 + X 3 + X 4 ). 18 Suppose that – in any given time period – a certain stock is equally likely to go up 1 unit or down 1 unit, and that the outcomes of different periods are independent.

These continuous random variables can take on any value within some interval. For example, such random variables as the time it takes to complete an assignment, or the weight of a randomly chosen individual, are usually considered to be continuous. Every continuous random variable X has a function f associated with it. This function, called the probability density function of X, determines the probabilities associated with X in the following manner. For any numbers a < b, the area under f between a and b is equal to the probability that X assumes a value between a and b.

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