ECON 240ALast Tuesday & Lab TwoThe Normal DistributionOutlinePowerPoint PresentationProbability Density FunctionCumulative Distribution FunctionSlide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15For the Binomial DistributionSlide 17Normal Approximation to the binomialSlide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Normal Approximation to the Binomial: De MoivreGuidelines for using the normal approximationThe Standardized Normal VariateSlide 29Slide 30Normal Variate xSlide 32Slide 33Slide 34For the Normal DistributionSlide 36Sample MeansSample Mean ExampleRate of Return UC Stock Index Fund, http://atyourservice.ucop.edu/Table Cont.Slide 41Slide 42Slide 43Slide 44Data ConsiderationsCont.Slide 47What are the properties of this sample mean?Slide 49Properties ofCentral Limit TheoremSlide 52The rate of return, ri , could be distributed as uniformAnd yet for a large sample, the sample mean will be distributed as normalBottom LineSlide 56Sample Standard DeviationSlide 58Slide 591ECON 240AECON 240APower 5Power 52Last Tuesday & Lab TwoLast Tuesday & Lab TwoProbabilityProbabilityDiscrete Binomial Probability Discrete Binomial Probability DistributionDistribution3The Normal DistributionThe Normal Distribution4OutlineOutlineThe cumulative distribution function, The cumulative distribution function, or sigmoid (S-shaped curve)or sigmoid (S-shaped curve)The normal distribution as an The normal distribution as an approximation to the binomialapproximation to the binomialThe standardized normal variable, zThe standardized normal variable, zsample meanssample meansThe distribution of the sample The distribution of the sample mean: the normal distributionmean: the normal distributionBinomial DistributionFive Flips of a Fair Coindensity cumulativek n p0 5 0.03125 0.031251 5 0.15625 0.18752 5 0.3125 0.53 5 0.3125 0.81254 5 0.15625 0.968755 5 0.03125 16Probability Density Probability Density FunctionFunctionFive Flips of a Fair Coin00.050.10.150.20.250.30.350 1 2 3 4 5Number of HeadsProbability7Cumulative Distribution Cumulative Distribution FunctionFunctionFive Flips of a Fair Coin00.10.20.30.40.50.60.70.80.910 1 2 3 4 5Number of HeadsProbability8Cumulative Distribution Cumulative Distribution FunctionFunctionThe probability of getting two or less The probability of getting two or less heads in five flips is 0.5heads in five flips is 0.5•can use the cumulative distribution functioncan use the cumulative distribution function•can use the probability density function and can use the probability density function and add the probabilities for 0, 1, and 2 headsadd the probabilities for 0, 1, and 2 headsthe probability of getting two heads or the probability of getting two heads or three heads isthree heads is•can add the probabilities for 2 heads and three can add the probabilities for 2 heads and three heads from the probability density function heads from the probability density function9Cumulative Distribution Cumulative Distribution FunctionFunctionFive Flips of a Fair Coin00.10.20.30.40.50.60.70.80.910 1 2 3 4 5Number of HeadsProbabilityBinomial DistributionFive Flips of a Fair Coindensity cumulativek n p0 5 0.03125 0.031251 5 0.15625 0.18752 5 0.3125 0.53 5 0.3125 0.81254 5 0.15625 0.968755 5 0.03125 111Cumulative Distribution Cumulative Distribution FunctionFunctionthe probability of getting two heads or the probability of getting two heads or three heads is:three heads is:•can add the probabilities for 2 heads and can add the probabilities for 2 heads and three heads from the probability density three heads from the probability density function function •can use the probability of getting up to 3 can use the probability of getting up to 3 heads, P(3 or less heads) from the heads, P(3 or less heads) from the cumulative distribution function (CDF) and cumulative distribution function (CDF) and subtract the probability of getting up to one subtract the probability of getting up to one head P(1 or less heads]head P(1 or less heads]12Probability Density Probability Density FunctionFunctionFive Flips of a Fair Coin00.050.10.150.20.250.30.350 1 2 3 4 5Number of HeadsProbabilityBinomial DistributionFive Flips of a Fair Coindensity cumulativek n p0 5 0.03125 0.031251 5 0.15625 0.18752 5 0.3125 0.53 5 0.3125 0.81254 5 0.15625 0.968755 5 0.03125 1Binomial DistributionFive Flips of a Fair Coindensity cumulativek n p0 5 0.03125 0.031251 5 0.15625 0.18752 5 0.3125 0.53 5 0.3125 0.81254 5 0.15625 0.968755 5 0.03125 115Cumulative Distribution Cumulative Distribution FunctionFunctionFive Flips of a Fair Coin00.10.20.30.40.50.60.70.80.910 1 2 3 4 5Number of HeadsProbability]3[ hp]1[ hp16For the Binomial For the Binomial DistributionDistributionCan use a computer as we did in Can use a computer as we did in Lab TwoLab TwoCan use Tables for the cumulative Can use Tables for the cumulative distribution function of the distribution function of the binomial such as Table 1 in the text binomial such as Table 1 in the text in Appendix B, p. B-1in Appendix B, p. B-1•need a table for each p and n.need a table for each p and n.1718Normal Approximation to Normal Approximation to the binomialthe binomialFortunately, for large samples, we Fortunately, for large samples, we can approximate the binomial with can approximate the binomial with the normal distribution, as we saw the normal distribution, as we saw in Lab Twoin Lab TwoForty Tosses of a Fair Coin00.020.040.060.080.10.120.14036912151821242730333639Number of HeadsProbabiltyBinomial Probability Density FunctionForty Tosses of a Fair Coin00.10.20.30.40.50.60.70.80.91036912151821242730333639Number of Heads ProbabilityBinomial Cumulative Distribution Function21The Normal DistributionThe Normal DistributionWhat would the normal density What would the normal density function look like if it had the same function look like if it had the same expected value and the same expected value and the same variance as this binomial variance as this binomial distributiondistribution•from Power 4, E(h) = n*p =40*1/2=20from Power 4, E(h) = n*p =40*1/2=20•from Power 4, VAR[h] = n*p*(1-p) = from Power 4, VAR[h] = n*p*(1-p) = 40*1/2*1/2 =1040*1/2*1/2 =10Normal Density Function, Mean 20, Variance 1000.020.040.060.080.10.120.140 10 20 30 40 50Number of HeadsDensityComparing the Normal Density with the Binomial Probability Distribution00.020.040.060.080.10.120.14036912151821242730333639Number of
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