By William Feller
Compatible for self research Use actual examples and actual information units that would be time-honored to the viewers creation to the bootstrap is incorporated – it is a glossy approach lacking in lots of different books
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Additional resources for An introduction to probability theory and its applications, vol. 2
1O"Pij,t=O .. n,n=I .. 8)],insequence=true); 3. Find the critical points for each of the following equations. Plot a few trajectories to confirm the locations of the basins of attractions. a. ¥Ii -y(t)(1 - yet»~. = > solve(y"(1-y)=O,y); > with(DEtools): > DEplot1 (diff(y(t),t)=-y(t)*(1-y(t»,y(t), t=O .. 2); b. 4)}: > DEplot2(eqns,[x,Y),t=O .. 4,inits,x=-I .. S,y=-1 .. 5); 47 Chapter 2 I Some Mathematical Tools = = AZ(t), Z(O) C, with A a constant square matrix and C a vector is exp(At)C. Compute this exponential in the case 4.
Regression Principle. The method of least squares can be adapted to calculate the parameters of a model if there is some transformation of the model that is linear in the transformed parameters. Consider the Michaelis-Menten equation for the initial reaction rate Vo of the enzyme-catalyzed reaction of a substrate having a concentration denoted by [S], vmax[S] Vo = Km + [S]; the parameters are Vmax and Km. By reciprocating both sides of this equation we get the Lineweaver-Burk equation I Km I Vo - Vmax [S] I vmax' -----+-- Now the transformed model is linear in its parameters M = Kmlvmax and B = Ilvmax and the transformed data are Y = IlvoandX = I/[S].
_ ,. - " \. 3) Initial value problems call be solved numerically. 4) specifies the slope of the solution required by the differential equation at every point (t, y) in the domain. This may be visualized by plotting a short line segment having that slope at each point. 3). Such a plot is called a direc(iollfield. Solutions to the equation must follow the field and cannot cross slopes. With such a direction field it is possible to sketch solutions manually.
An introduction to probability theory and its applications, vol. 2 by William Feller