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

By Sheldon M. Ross

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This result is easily proven by using the approximating processes, as we now show. 1 Given that X (t) = x, the conditional probability law of the collection of prices X (y), 0 ≤ y ≤ t, is the same for all values of μ. Proof. Let s = X (0) be the price at time 0. Now, consider the approximating model where the price changes √ every time units by an amount equal, in absolute value, to c ≡ σ , and note that c does not depend on μ. By time t, there would have been t/ changes. Hence, given that the price has increased from time 0 to time t by the amount x − s, it follows that, of the t/ changes, there have been a total of 2t + x−s 2c positive changes and a total of 2t − x−s negative changes.

Properties of Normal Random Variables An important property of normal random variables is that if X is a normal random variable then so is aX +b, when a and b are constants. This property enables us to transform any normal random variable X into a standard normal random variable. For suppose X is normal with mean μ and variance σ 2 . 8)) Z= X −μ σ Properties of Normal Random Variables 27 has expected value 0 and variance 1, it follows that Z is a standard normal random variable. As a result, we can compute probabilities for any normal random variable in terms of the standard normal distribution function .

If a member of the club is randomly chosen, what is the conditional probability that she (a) plays chess given that she plays bridge; (b) plays bridge given that she plays chess? 5 Cystic fibrosis (CF) is a genetically caused disease. A child that receives a CF gene from each of its parents will develop the disease either as a teenager or before, and will not live to adulthood. A child that receives either zero or one CF gene will not develop the disease. If an individual has a CF gene, then each of his or her children will independently receive that gene with probability 1/2.

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