C HAPTER 7 Sum of Random Variables and Long-term Averages

1. CHAPTER 7 Sum of Random Variables and Long-term Averages Prof. Sang-Jo Yoo sjyoo@inha.ac.kr http://multinet.inha.ac.kr

2. What we are going to study? • Motivation • Many problems involve the counting of the number of occurrences of events or computation of arithmetic averages in a series of measurements. • These problems can be reduce to the problem of finding the distribution of a random variable consisting of the sum of n independent, identically distributed (iid) random variables. • Contents • PDF of the sum of independent random variables • Sample mean estimator • Central limit theorem • The CDF of a sum of random variables approaches that of a Gaussian random variable even through the CDF of individual random variables may be far from Gaussian. • This result enables us to approximate the PDF of sums of random variables by the PDF of a Gaussian random variable.

3. Sum of Random Variables • Mean and variance of sum of random variables • Let be a sequence of random variables, and let • Then, • Regardless of statistical dependence, the expected value of a sum of n random variables is equal to the sum of the expected values • Variance • Example 7.1 • Generalized  • In general, the variance of a sum of RVs is not equal to the sum of the individual variables. • If are independent RVs, then • Example 7.2

4. PDF of Sums of Independent RVs • Let be n independent random variables, to find the PDF of , we are going to use transform methods. • Let and (n=2 case) are independent random variables, and • Characteristic function of Z is given by • Recall: can be viewed as the Fourier transform of the PDF of Z.

5. PDF of Sums of Independent RVs • Let be independent random variables, and • To find the PDF of Sn, originally we need to compute  very complex • We can use the transform method • For integer-valued random variables, the probability generating function for a sum of independent RV's is preferred. Example 7.3 Example 7.4

6. Sample Mean and the Law of Large Numbers • Let be a RV with . Let , denote independent and repeated measurements of . • are iid RV's with the same pdf as • The sample mean of the sequence is used to estimate • The expected value of sample mean • The sample mean is equal to on the average.

7. Sample Mean • The mean square error of the sample mean about is equal to the variance of Mn, • Variance of Mn • Note that Mn=Sn/n, where Sn=X1+X2+…+Xn.  • Since Xj’s are iid random variables, • The probability that the sample mean is close to the true mean approaches one as becomes very large. Note 7-1

8. Example 7.9 • A voltage of constant, but unknown, value is to be measured. Each measurement is actually the sum of the desired voltage and a noise voltage of zero mean and standard deviation of 1V. • Assume that the noise voltage are independent random variables. How many measurements are required so that the probability that is within of the true mean is at least 0.99?

9. Central Limit Theorem • Recall: Let be the sum of iid random variables with finite mean and finite variance . • Then the PDF of can be obtained from • Central Limit Theorem: as n becomes large, the CDF of a properly normalizedSn approaches that of a Gaussian random variable. • Let be the zero-mean , unit-variance random variable defined by • then Note 7-2

10. Example 7.11, Example 7.12 (Note 7-3) Suppose that orders at a restaurant are iid random variables with mean and standard deviation . 1) Estimate the probability that the first 100 customers spend a total of more than $840. 2) Estimate the probability that the first 100 customers spend a total of between $780 and $820. 3) After how many orders can we be 90% sure that the total spent by all customers is more than $1000?