Probability and Statistical Inference (9th Edition) Chapter 5 (Part 2/2)

1. Probability and Statistical Inference (9th Edition)Chapter 5 (Part 2/2) Distributions of Functions of Random Variables November 25, 2015

2. Outline 5.5 Random Functions Associated with Normal Distributions 5.6 The Central Limit Theorem 5.7 Approximations for Discrete Distributions 5.8 Chebyshev’s Inequality and Convergence in Probability

3. Random Functions Associated with Normal Distributions • Theorem: Assume that X1, X2,…, Xnare independent random variables with distributions N(μ1,σ12), N(μ2,σ22),…, N(μn,σn2), respectively. Then,

4. Random Functions Associated with Normal Distributions • Proof: Recall the mgf of N(m,s2) is

5. Random Functions Associated with Normal Distributions • Example 1: If X1 and X2 are independent normal random variables N(µ1,σ12) and N(µ2,σ22), respectively, then X1 + X2 is N(µ1+µ2, σ12+σ22), and X1 - X2 is N(µ1-µ2, σ12+σ22)

6. Random Functions Associated with Normal Distributions • Example 2: If X1,X2,…,Xn correspond to random samples from a normal distribution N(μ,σ2), then the sample mean is N(μ,σ2/n) • Proof:

7. Random Functions Associated with Normal Distributions • One important implication of the distribution of is that it has a greater probability of falling in an interval containing μ than does a single sample Xk • The larger the sample size n, the smaller the variance of the sample mean • “Mean” is a constant, but “sample mean” is a random variable

8. Random Functions Associated with Normal Distributions • For example, assume that X1, X2,…, Xn are random samples from N(50,16) distribution. Then, is N(50,16/n). The following figure shows the pdf of with different values of n

9. Random Functions Associated with Normal Distributions

10. Random Functions Associated with Normal Distributions • Recall: • Let Z1, Z2, …, Zn be i.i.d. N(0,1). Then, w = Z12+ Z22+ …+ Zn2 is χ2(n) • Let X1, X2,…, Xn be independent chi-square random variables with k1, k2,…, kn degrees of freedom, i.e., χ2(k1), χ2(k2),…, χ2(kn), respectively. Then, Y=X1+X2+…+Xn is χ2(k1+k2+…+kn)

11. Random Functions Associated with Normal Distributions • Theorem: Let X1, X2,…, Xn be random samples from the N(μ,σ2) distribution. The sample mean and sample variance are given by Then, (a) and are independent (b) is χ2(n-1)

12. Random Functions Associated with Normal Distributions • We will accept (a) without proving it • Proof of (b):

13. Random Functions Associated with Normal Distributions

14. Random Functions Associated with Normal Distributions

15. Random Functions Associated with Normal Distributions • It is interesting to observe that • That is, when the actual mean is replaced by the sample mean, one degree of freedom is lost

16. Central Limit Theorem • It is useful to first review some related theorems • Theorem (Sample Mean): Let X1, X2, …, Xn be a sequence of i.i.d. random variables with mean m and variance s2. Then, the sample mean is a random variable with mean m and variance s2/n

17. Central Limit Theorem • Theorem (Strong Law of Large Numbers): Let X1, X2, …, Xn be a sequence of i.i.d. random variables with mean m. Then, with probability 1, That is, (The sample mean converges almost surely, or converges with probability 1, to the expected value)

18. Central Limit Theorem • Theorem (Strong Law of Large Numbers)(Cont.): • This theorem holds for any distribution of the Xi’s • This is one of the most well-known results in probability theory

19. Central Limit Theorem • Theorem (Central Limit Theorem): Let X1, X2, …, Xn be a sequence of i.i.d. random variables with mean m and variance s2. Then the distribution of is N(0,1) as That is, (convergence in distribution)

20. Central Limit Theorem • Theorem (Central Limit Theorem)(Cont.): • While tends to “degenerate” to zero (Strong Law of Large Numbers), the factor in “spreads out” the probability enough to prevent this degeneration

21. Central Limit Theorem • Theorem (Central Limit Theorem)(Cont.): • One observation that helps make sense of this result is that, in the case of normal distribution (i.e., X1, X2, …, Xn are i.i.d. normal), is N(m,s2/n) • Hence, is (exactly) N(0,1) for each positive value of n • Thus, in the limit, the distribution must also be N(0,1)

22. Central Limit Theorem • Theorem (Central Limit Theorem)(Cont.): • The powerful fact is that this theorem holds for any distribution of the Xi’s • It explains the remarkable fact that the empirical frequencies of so many natural “populations” exhibit a bell-shaped (i.e., normal) curve • The term “central limit theorem” traces back to George Polya who first used the term in 1920 in the title of a paper. Polya referred to the theorem as “central” due to its importance in probability theory

23. Central Limit Theorem • The Central Limit Theorem and the Strong Law of Large Numbers are the two fundamental theorems of probability

24. Central Limit Theorem • Example 1 (Normal Approximation to the Uniform Sum Distribution (a.k.a. the Irwin-Hall Distribution)): Let Xi, i=1,2,… be i.i.d. U(0,1). Compare the graph of the pdf of Y=X1+X2+…+Xn, with the graph of the N(n(1/2), n(1/12)) pdf n=2

25. Central Limit Theorem n=4

26. Central Limit Theorem • Example 2 (Normal Approximation to the Uniform Sum Distribution (a.k.a. the Irwin-Hall Distribution)): Let Xi, i=1,2,…,10 be i.i.d. U(0,1). Estimate P(X1+X2+…+X10 > 7) • Solution: With and by the central limit theorem,

27. Central Limit Theorem • Example 3 (Normal Approximation to the Chi-Square Distribution): Let X1,X2,…,Xn be i.i.d. N(0,1). Then, is chi-square with n degrees of freedom, with E(Y)=n and Var(Y)=2n • Recall the pdf of Y is • Let

28. Central Limit Theorem • The pdf of W is given by • Compare the pdf of W and the pdf of N(0,1): n=100 n=20

29. Approximations for Discrete Distributions • The beauty of the central limit theorem is that it holds regardless of the underlying distribution (even discrete)

30. Approximations for Discrete Distributions • Example 4 (Normal Approximation to the Binomial Distribution): X1,X2,… Xn are random samples from a Bernoulli distribution with μ=p and σ2 = p(1-p). Then, Y=X1+X2+…+Xn is binomial b(n,p). The central limit theorem states that is N(0,1) as n approaches infinity

31. Approximations for Discrete Distributions • Thus, if n is sufficiently large, the distribution of Y is approximately N(np,np(1-p)), and the probabilities for the binomial distribution b(n,p) can be approximated with this normal distribution, i.e., for sufficiently large n

32. Approximations for Discrete Distributions • Consider n=10, p=1/2, i.e., Y~b(10,1/2). Then, by CLT, Y can be approximated by the normal distribution with mean 10(1/2)=5 and variance 10(1/2)(1/2)=5/2. Compare the pmf of Y and the pdf of N(5,5/2):

33. Approximations for Discrete Distributions • Example 5 (Normal Approximation to the Poisson Distribution): • Recall the Poisson pmf where parameter is both the mean and variance of the distribution • Poisson random variable counts the number of discrete occurrences (sometimes called “events” or “arrivals”) that take place during a time-interval of given length

34. Approximations for Discrete Distributions • A random variable having a Poisson distribution with mean 20 can be thought of as the sum Y of the observations of a random sample of size 20 from a Poisson distribution with mean 1. Thus, has a distribution that is approximately N(0,1), and the distribution of Y is approximately N(20,20)

35. Approximations for Discrete Distributions • Compare the pmf of Y and the pdf of N(20,20):

36. Markov’s Inequality • Theorem (Markov’s Inequality): If X is a continuous random variable that takes only nonnegative values, then for any a>0, • The inequality is valid for all distributions (discrete or continuous)

37. Markov’s Inequality • Proof:

38. Markov’s Inequality • Intuition behind Markov’s Inequality, using a fair dice (discrete) example: Then, …

39. Chebyshev’s Inequality • Theorem (Chebyshev’s Inequality): If X is a continuous random variable with mean m and variance s2, then for any k>0, • The inequality is valid for all distributions (discrete or continuous) for which the standard deviation exists

40. Chebyshev’s Inequality • Proof: Since (X-m)2 is a nonnegative random variable, we can apply Markov’s inequality (with a=k2) to obtain Thus,

41. Chebyshev’s Inequality (Another Form) • Chebyshev’s Inequality (another form): • Chebyshev’s inequality states that the probability that X differs from its mean by at least k standard deviations is less than or equal to 1/k2 • It follows that the probability that X differs from its mean by less than k standard deviations is at least 1-1/k2

42. Chebyshev’s Inequality • The importance of Markov’s and Chebyshev’s inequalities is that they enable us to derive (sometimes loose but still useful) bounds on probabilities when only the mean, or both the mean and the variance, of the probability distribution are known

43. Chebyshev’s Inequality • Example 1: If it is known that X has a mean of 25 and a variance of 16, then, a lower bound for P(17<X<33) is given by and an upper bound for P(|X-25|>=12) is • The results hold for any distribution with mean 25 and variance 16