I. IntroductionIn the case of a large number of trials, the binomial distribution can be approximated by the normal distribution so that we can deal with one table of numbers rather than many tables that depend on the parameters of the binomial, n and p. The normal distribution is a continuous distribution and we examine continuous random variables, their central tendency, and their dispersion, in general. As an example, we explore the uniform distribution in addition to the normal distribution.II. The Normal Approximation to the BinomialIII. Continuous VariablesIV. The Sampling Distribution of the MeanDateRate of Return, UC Index FundV. Student’s t-DistributionVI. SamplingOct. 7, 2004 LEC #5 ECON 240A-1 L. PhillipsNormal DistributionI. IntroductionIn the case of a large number of trials, the binomial distribution can be approximated by the normal distribution so that we can deal with one table of numbers rather than many tables that depend on the parameters of the binomial, n and p. The normal distribution is a continuous distribution and we examine continuous random variables, their central tendency, and their dispersion, in general. As an example, we explore the uniform distribution in addition to the normal distribution.In terms of applications, we need to move beyond estimating proportions to estimating sample means. We have seen how important independence is to the process of calculating the expected value and the variance of sums of random variables. A random sample, with each member chosen independently, is the key. We will look at the issue of sampling.II. The Normal Approximation to the BinomialRecall from Figures 2, 3, and 6 in the previous lecture that the distribution (histogram) of the binomial distribution depends on the number of trials. In general, this discrete probability distribution varies directly with the parameters n and p. In the days before desktop computers, this was a particular pain and users had to refer to tables of thebinomial distribution to calculate, for example, the probability of obtaining less than seven heads in ten flips of a fair coin. Abraham De Moivre showed that for a large number of trials, the binomial distribution could be approximated by a single continuous distribution, the normal. Amazingly perhaps, this works for any value of p. The approximation is:P(a  k  b)  P[(a – ½ - np)/)1( pnp  z  [(b + ½ - np)/)1( pnp ]Oct. 7, 2004 LEC #5 ECON 240A-2 L. PhillipsNormal DistributionWhere z is the standard normal variate with mean zero and variance one, and ½ is called the continuity correction for the smooth density approximation of z to a discrete histogram for k. As a rule of thumb, this is a good approximation when np  5 and n (1 – p)  5.The continuous density function for the standardized normal variate, z is:f (z) = [1/2 ] exp{ -1/2 [(z – 0)/1]2 }III. Continuous VariablesThe standardized normal variate z ranges from minus infinity to plus infinity along the number line. Any normal variable, x, with mean, , and variance, 2 , can be expressed in terms of z:z = (x - )/ , or rearranging,x =  +  z , so x is just a linear function of z, with density function, f(x),f(x) = (1/2) exp {-1/2[(x - )/]2 }.In general, the density function of a continuous random variable, y, is f(y). Another example is the uniform distribution, where the variable u ranges from zero to one, with probability density equal to one in this range, as illustrated in Figure 1: DensityfunctionOct. 7, 2004 LEC #5 ECON 240A-3 L. PhillipsNormal DistributionFigure 1: Density function for the Continuous Uniform Variate---------------------------------------------------------------------------------------------------For a continuous random variable, y, its expected value is:E (y) = dyyyf )(.For the uniform variable, 0  u  1, the expected value is:E(u) = 10)( duuuf= 10*1* duu= u2/2 01 =1/2 .For a continuous variable y, its variance is:VAR(y) = E(y –Ey)2 =E[ y2 –2yEy + (Ey)2 ] = Ey2 – (Ey)2 , i.e The second moment minus the square of the first moment. In terms of the density function,VAR(y) = [y – Ey]2 f(y) dy = [y2 –2yEy + (Ey)2 ] f(y) dy0Uniform variate01Oct. 7, 2004 LEC #5 ECON 240A-4 L. PhillipsNormal Distribution= y2 f(y) dy –2Ey y f(y) dy + (Ey)2 f(y) dy= y2 f(y) dy – [ y f(y) dy ]2 .In the case of the uniform variable u,VAR( u ) = E[u – Eu]2 = E u2 - (1/2)2 = 10u2 f(u) du –1/4= (u3 /3)01 - ¼ = 1/3 – ¼ = 1/12.The probability of finding a continuous random variable y with value less than or equal to b is the cumulative distribution function, F(b):P(y  b) = bf(y) dy = F(y) b= F(b) – F( -) = F(b) – 0 = F(b).Note that the probability that y is exactly b is equal to zero:P(b  y  b) = bbf(y) dy = F(b) – F(b) =0.In the case of the uniform distribution, the probability that u is less than or equal to u* is:P( u  u* ) = *0uf(u) du = u 0*u = u* = F(u*), as illustrated in Figure 2: Probability, F(u)1Oct. 7, 2004 LEC #5 ECON 240A-5 L. PhillipsNormal DistributionFigure 2: Cumulative distribution Function for Continuous Random Variable------------------------------------------------------------------------------------------ IV. The Sampling Distribution of the MeanSuppose we examine the monthly rate of return for investment funds. The monthly rate of return, ri , for asset i is the capital gain (loss), or change in asset price, p(t)– p(t-1), plus dividends, D(t), relative to the previous period’s price p(t-1):ri = [ p(t) – p(t-1) + D(t)]/ p(t-1) .The price of the asset this period, p(t), is highly correlated with the price last period, p(t-1), but the change in price, p(t) – p(t-1), is not correlated with the change from the previous period, p(t-1) – p(t-2). As a consequence, the rate of return on asset i this period,r(t), tends to be independent of the rate of return from the previous period, r(t-1), so that we have the property of independence for a sequence of monthly rates of return. Thus a sample of twelve monthly rates of return on an asset, for example, will satisfy the requirement of independence as if the sample had been selected randomly.The rate of return for the last twelve months of the stock index fund open to investment by UC employees is presented in the following table. It is available at the URL: