Oct 10 2002 LEC 5 ECON 240A 1 Normal Distribution L Phillips I Introduction In 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 Binomial Recall 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 the binomial 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 np np 1 p z b np np 1 p Oct 10 2002 LEC 5 ECON 240A 2 Normal Distribution L Phillips Where z is the standard normal variate with mean zero and variance one and 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 Variables The 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 Density function Oct 10 2002 LEC 5 ECON 240A 3 Normal Distribution L Phillips 0 1 0 Uniform variate Figure 1 Density function for the Continuous Uniform Variate For a continuous random variable y its expected value is E y yf y dy For the uniform variable 0 u 1 the expected value is 1 E u uf u du 0 u2 2 1 0 1 u 1 du 0 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 dy Oct 10 2002 LEC 5 ECON 240A 4 Normal Distribution L Phillips 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 1 u2 f u du 1 4 0 1 0 u3 3 1 3 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 b P y b f 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 b f y dy F b F b 0 P b y b b In the case of the uniform distribution the probability that u is less than or equal to u is u P u u f u du u 0 as illustrated in Figure 2 1 Probability F u u 0 u F u Oct 10 2002 LEC 5 ECON 240A 5 Normal Distribution L Phillips 45 degrees 0 0 Uniform variable u 1 Figure 2 Cumulative distribution Function for Continuous Random Variable IV The Sampling Distribution of the Mean Suppose 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 http www ucop edu bencom rs perform html Oct 10 2002 LEC 5 ECON 240A 6 Normal Distribution L Phillips Table 1 Monthly Rate of Return UC Stock Index Fund Date August 99 September 99 October 99 November 99 December 99 January 2000 February 2000 March 2000 April 2000 May 2000 June 2000 July 2000 Rate of Return UC Index Fund 2 46 2 44 7 48 3 79 5 48 1 95 2 67 8 78 1 45 0 56 1 97 2 03 The average rate of return over the past twelve months r for the stock index fund is July 2000 r r j 12 1 61 j Aug 99 If mu is the mean of the distribution of the rate of return for the stock index fund and assuming it does not change over time which may not be true or realistic then July 2000 E r E r j 12 12 12 j Aug 99 If 2 is the …
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