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Chapter6_2

Chapter6_2 - Random Variables Definitions Chapter 6...

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1 Chapter 6: Probability Distributions Read Chapter 7 Random Variables Definitions A Random Variable , X , is a real valued function whose values are determined by a random experiment. Mean: E(X)= μ = Σ x p(x) all x Variance: E[(X- μ ) 2 ]= σ 2 = Σ (x- μ ) 2 p(x) all x Mass function: P(X=x) = p(x) Standard Deviation = σ The Normal Distribution The normal distribution is often called the Gaussian distribution, after Carl Friedrich Gauss, who discovered many of its properties. Gauss, commonly viewed as one of the greatest mathematicians of all time (if not the greatest), was honored by Germany on their 10 Deutschmark bill. The Gaussian distribution is the probabilistic model for much of this course. 2 2 2 ) ( 2 2 2 1 ) , | ( σ μ πσ σ μ = x e x f Normal Distribution Normal distributions are “Bell shaped” Symmetric around the mean The mean ( μ ) and the standard deviation ( σ ) completely describe the density curve Increasing/decreasing μ moves the curve along the horizontal axis Increasing/decreasing σ controls the spread of the curve xx Density 100 200 300 400 500 0.0 0.02 0.04 0.06 0.08 Different Means and Variances Z-Scores and the Standard Normal Distribution The z-score for a value x of a random variable is the number of standard deviations that x falls from the mean A negative (positive) z-score indicates that the value is below (above) the mean z-scores can be used to calculate the probabilities of a normal random variable using the normal tables in the back of the book z = x μ σ

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2 Z-Scores and the Standard Normal Distribution A standard normal distribution has mean μ=0 and standard deviation σ =1 When a random variable has a normal distribution and its values are converted to z -scores by subtracting the mean and dividing by the standard deviation, the z - scores have the standard normal distribution. Table A: Standard Normal Probabilities Table A enables us to find normal probabilities It tabulates the normal cumulative probabilities falling below the point μ +z σ To use the table: Find the corresponding z -score Look up the closest standardized score ( z ) in the table. First column gives z to the first decimal place First row gives the second decimal place of z The corresponding probability found in the body of the table gives the probability of falling below the z -score Using Table A Find the probability that a normal random variable takes a value less than 1.43 standard deviations above μ; P(Z<1.43)=.9236 Stat Crunch to the Rescue!! Using Table A Find the probability that a normal random variable takes a value greater than 1.43 standard deviations above μ: P(Z>1.43)=1-.9236=.0764
3 Find the probability that a normal random variable assumes a value within 1.43 standard deviations of μ Probability below 1.43 σ = .9236 Probability below -1.43 σ = .0764 (=1-.9236) P(-1.43<Z<1.43) =.9236-.0764=.8472 P(-1.43<Z<1.43) = .9236-.0764 = .8472 How Can We Find the Value of z for a Certain Cumulative Probability?

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