Mathematical Analisys (second course) and Probability A.A. 2000/2001 - Prof. G. Stefani 1. Sequences and series in

Mathematical Analisys (second course) and Probability A.A. 2000/2001 - Prof. G. Stefani

1. Sequences and series in R. Definitions. Conergence criterions. Cauchy product of two series.

2. Sequences and series of functions. Definition of pointwise and uniform convergence.

Uniform limit of continuous functions. Integrals, derivations and limits. Power series. Radious of convergence. Taylor series. Taylor expansion of main functions. Exponential map in C.

3. Differential Calculus in R

. Norm, open, closei, compact, connex sets. Limits and conti- nuity. Partial and directional derivatatives, differential. Higher order derivatives. C

functions and Taylor’s formula. Local and global maxima and minima. Stationary points. Necessary and sufficient conditions for local extrema. Implicit function theorem. Constrained estrema, Lagrange multipliers (only first order conditions). Functions with values in R

, Jacobi matrix, derivative of the composition of two functions.

4. Integrals in R

. Riemann Integral on bounded domins. Reduction formula for normal domins.

Change of coordinates. Baricenter.

5. Discret probability. Probabily spaces, Bayes formula, stocastic indipendence. Discret and independent random variables. Bernulli Scheme. Binomial, ipergeometric, geometric, Poisson dis- tributions. Limit of ipergeometric and binomial distributions. Expected value, variance, standard deviation, covariance and their propeties. Chebichev inequality.

6. Probability in R. Probabily spaces. Random variables: distribution function, density (only continuous but in a finite number of points). Gauss, esponential, gamma lows. Expected value, variance, standard deviation, covariance and their propeties. Chebichev inequality.

7. Approximation. Large numbers low and its meaning. Central limit theorem, its meaning and comparison with the large numbers low. Normal approximation.

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