Introducing to Matlab and it's Graphics Capabilities
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Uk+1 < p) p = (−1)k (log( p))k , 0 < p < 1. k! 12) In choosing p = e−λ , we get: U1 < e−λ U1 ≥ e−λ and U1 U2 < e−λ =⇒ X = 0, =⇒ X = 1, .. , =⇒ X = k. ⎪ U1 U2 . . Uk ≥ e−λ and ⎪ ⎪ ⎪ ⎭ U1 U2 . . 13) This algorithm gives us Prob(X = k) = e−λ λk . k! 15) is F (this was shown in Chapter 2). Now this result helps us to generate variables with known distribution functions. 16) and its cumulative distribution function is: FX (x) = 1 π x −∞ du 1 π = arctan(x) − = u. 17) Thus U = FX (X ) =⇒ X = FX−1 (U ) = tan πU + π .
We can deduce it from the joint cumulative distribution function F X,Y (x, y). In fact Prob(X ≤ x) = Prob(X ≤ x Y ≤ +∞). 80) This gives FX (x) = FX,Y (x, +∞). 81) The marginal probability density function of X is given by the derivative of the marginal cumulative distribution function: f X (x) = d FX (x). 83) then FX (x) = FX,Y (x, +∞) = x −∞ +∞ −∞ f X,Y (u, v)dudv. 85) from which f X (x) = +∞ −∞ f X,Y (x, v)dv. 86) 24 Stochastic Simulation and Applications in Finance For practical purposes, in order to eliminate a variable, all we need is to integrate the function from −∞ to +∞ with respect to the variable in question to obtain the marginal density.
M, transform into the same vector y. 170) Introduction to Random Variables 35 Let J be the Jacobian of the transformation deﬁned by ⎡ ∂y 1 ... ⎢ ∂ x1 ⎢ . J (x) = ⎢ ⎢ .. ⎣ ∂y . n ... ∂ x1 ∂ y1 ∂ xn .. ∂ yn ∂ xn ⎤ ⎥ ⎥ ⎥. 171) The joint density function of Y is given by: m f Y (y) = i=1 f X (x i ) . 1 Afﬁne transformation of a Gaussian vector Consider the Gaussian random vector X with mean m X and covariance matrix X , denoted by N (m X , X ). Set Y = X + μ, where is a matrix with appropriate dimensions.