Final Review Guide

Exercises p632 1 10 94 approximate solutions of

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* Exercises: p.632 1-10 9.4 Approximate Solutions of Differential Equations * Euler’s Method: Suppose we are given the differential equation dy dx = F ( x, y ) subject to the initial condition y ( x 0 ) = y 0 and we wish to find an approximation of y ( b ) where b is a number greater than x 0 and n is a positive integer. Compute h = b - x 0 n x 1 = x 0 + h x 2 = x 0 + 2 h x 3 = x 0 + 3 h · · · x n = x 0 + nh = b and 8
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y 0 = y ( x 0 ) y 1 = y 0 + hF ( x 0 , y 0 ) y 2 = y 1 + hF ( x 1 , y 1 ) . . . y n = y n - 1 + hF ( x n - 1 , y n - 1 ) Then, y n gives an approximation of the true value y ( b ) of teh solution to the initial value problem at x = b . * Exercises: p.640 1-15 Chapter 10 Probability and Calculus 10.1 Probability Distributions of Random Variables * Terms: experiment, outcomes, sample points, sample space, event, probability of the event, finite discrete random variable, discrete probability function, histogram, continuous random variable, probability density function, exponential density function, exponentially distributed, joint probability density function * Probabilty Density Function: A probability density function of a r.v. X in an interval I , where I may be bounded or unbounded, is a nonnegative function f having the property that the total area of the region under the graph of f in the interval I is equal to 1. The probability that an observed value of the random variable X lies in the interval [ a, b ] is given by P ( a X b ) = Z b a f ( x ) dx * For a continuous random variable X , P ( a X b ) = P ( a < X b ) = P ( a X < b ) = P ( a < X < b ). * A joint probability density function of the random variables X and Y on a region D is a nonnegative function f ( x, y ) having the property Z D Z f ( x, y ) dA = 1 The probability that the observed values of teh random variables X and Y lie in a region R D is given by P [( X, Y ) in R ] = Z R Z f ( x, y ) dA * Exercises: pp.654-655 1-42 10.2 Expected Value and Standard Deviation * Terms: expected value, variance, standard deviation * Expected Value of a Discrete Random Variable X : Let X denote a random variable that assumes the values x 1 , x 2 , . . . , x n with associated probabilities p 1 , p 2 , . . . , p n , respectively. Then the expected value of X, E [ X ], is given by E [ X ] = x 1 p 1 + x 2 p + 2 + · · · + x n p n * Expected Value of a Continuous Random Variable: Suppose the function f defined on the interval [ a, b ] is the probability density function associated with a continuous random variable X . Then, the expected value of X is E [ X ] = Z b a xf ( x ) dx * Variance of a Discrete Random Variable: Let X be the discrete random variable two points above, and let X have expected value E [ X ] = μ . Then the variance of the random variable X is Var( X ) = p 1 ( x 1 - μ ) 2 + p 2 ( x 2 - μ ) 2 + · · · p n ( x n - μ ) 2 * Variance of a Continuous Random Variable: Let X be a continuous random variable with probability density funciton f ( x ) on [ a, b ] and expected value E [ X ] = μ . Then the variance of X is Var( X ) = Z b a ( x - μ ) 2 f ( x ) dx 9
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* Standard Deviation: For either a discrete or continuous random variable X , the standard deviation of X , denoted σ , is given by σ = p Var( X ) * Alternative Formula for Variance: For a continuos random variable X with probability density function f ( x ) defined on [ a, b ] and expected value E [ X ] = μ , we have Var( X ) = Z b a x 2 f ( x ) dx !
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