Read e-book online An Elementary Introduction to Mathematical Finance, Third PDF

By Sheldon M. Ross

ISBN-10: 0521192536

ISBN-13: 9780521192538

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Example text

4a Let X and Y both be Bernoulli random variables. That is, each takes on either the value 0 or 1. Using the identity Cov(X, Y ) = E[XY ] − E[X ] E[Y ] and noting that XY will equal 1 or 0 depending upon whether both X and Y are equal to 1, we obtain that Cov(X, Y ) = P{X = 1, Y = 1} − P{X = 1}P{Y = 1}. Covariance and Correlation 15 From this, we see that Cov(X, Y ) > 0 ⇐⇒ P{X = 1, Y = 1} > P{X = 1}P{Y = 1} P{X = 1, Y = 1} > P{Y = 1} P{X = 1} ⇐⇒ P{Y = 1 | X = 1} > P{Y = 1}. ⇐⇒ That is, the covariance of X and Y is positive if the outcome that X = 1 makes it more likely that Y = 1 (which, as is easily seen, also implies the reverse).

X n would be independent if each selected ball is replaced before the next selection is made, but they would not be independent if each selection is made without replacing previously selected balls. ) Whereas the average of the possible values of X is indicated by its expected value, its spread is measured by its variance. Definition The variance of X, denoted by Var(X ), is defined by Var(X ) = E[(X − E[X ])2 ]. Random Variables and Expected Values 13 In other words, the variance measures the average square of the difference between X and its expected value.

2 2 A plot of the normal probability density function gives a bell-shaped curve that is symmetric about the value μ, and with a variability that is measured by σ. The larger the value of σ, the more spread there is in f. 2 presents three different normal probability density functions. Note how the curve flattens out as σ increases. 2: Three Normal Probability Density Functions It can be shown that the parameters μ and σ 2 are equal to the expected value and to the variance of X, respectively. That is, μ = E[X ], σ 2 = Var(X ).

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


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