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Lecture 17

Lecture 17,18 Summary.pdf

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Michael Krashinsky

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Post-Lecture Seventeen and Eighteen ▯ De▯nition: A collection of random variables X1;X 2:::;X in called a sample if they are independent and identically distributed. ▯ Denote by X the sample mean of a sample X ;X ;:::;X and de▯ne it to 1 2 n be X = X 1 X + 2▯▯ + X n n ▯ By virtue of being a function of unpredictable quantities, the sample mean is also unpredictable, ie the sample mean is itself a random variable. ▯ Fact: If X is the sample mean of a sample X ;X ;:::;X having common 1 2 n distribution with mean ▯ apd standard deviation ▯ then X has mean ▯ also but standard deviation ▯= n ▯ Proving this is a simple application of our rules for mean and variances. 1 ▯ The Law of Large Numbers says that the sample mean would approach the distribution mean ▯ could you increase the sample size without bound; ie asymptotically the sample mean behaves like a constant. p ▯ This is easily understood by noting ▯ X▯ = ▯ together with ▯ = ▯X n which approaches zero as n increases without bound. ▯ Example. Consider the following discrete probability distribution with mean ▯ = 0:9. mean equals 0.9 0.5 0.4 0.3 P(X0.2 0.1 0.0 0 1 2 3 4 k ▯ I sampled from this distribution 500 times. { The ▯rst sample was of size 1 and a sample mean calculated. { The second sample was of size 2 and a sample mean calculated. . . { The last sample was of size 500 and a sample mean calculated. ▯ The (partial) results are tabled below. Sample Observed Values Sample Mean X x = 1 ▯ = 1 1 1 X 1X 2 x 1 2;x = 21 ▯ = 1:5 X ;X ;X x = 1;x = 4;x = 0 ▯ = 1:67 1 2 3 1 2 3 . . . X 1:::;X 498 x 1 0;x = 2;:::;x 498= 0 ▯ = 0:857 X ;:::;X x = 3;x = 0;:::;x = 1 ▯ = 0:892 1 499 1 2 499 X 1:::;X 500 x 1 0;x = 2;:::;x 500= 1 ▯ = ▯0:81 2 ▯ I plotted the sample means against the sample size. ▯ The plot illustrates a couple of things { For small n there is more variability in the sample mean and for large n there is less variability in the sample mean. { As the sample size n increases the sample mean x ▯ approaches ▯ = 0:9. 1.5 1.0 Sample Mean 0.5 0 100 200 300 400 500 Sample Size 3 Typical Textbook Question About the Law of Large Numbers. Exercise 4.91. An insurance company looks at the records for millions of homeowners and sees that the mean loss from ▯re in a year is ▯ = 300 dollars per person. The company plans to sell ▯re insurance for 300 dollars plus enough to cover its costs and pro▯t. 1. Explain clearly why it would be stupid to sell only 10 policies. Although the probability of having to pay for a total loss for one or more of the 10 policies is very small, if this were to happen, it would be ▯nancially disastrous. 2. Explain why selling thousands of such policies is a safe business. For thousands of policies, the law of large numbers says that the average claim on many policies will be close to the mean, so the insurance company can be assured that the premiums they collect will (almost certainly) cover the claims. 4 ▯ The Central Limit Theorem says that before the sample mean gets there (and for large n) the sample mean behaves like a random variable with the normal distribution with mean ▯ and standard deviation ▯= n; ie irrespective of the distribution that you’ve sampled from, the sample mean has approxi- mately the normal distribution (provided that the sample size is large.) ▯ n \large": depends on how non-normal the distribution you’ve sampled from is, the more non-normal it is, the bigger n has to be for the central limit theorem to kick in (as a general rule of thumb let’s say n > 30) ▯ Get this, If X1;X 2:::;X in a sample from a normal distribution itself, with mean ▯ and standard deviation ▯, then for any n (big or small) the sample mean X has exactly the normal distribution with mean ▯ and standard devi- p ation ▯= n; ie when sampling from the normal distribution itself, the Central Limit Theorem kicks in immediately. ▯ In Fact that result follows from the following important theorem: if 1 ;X2;:::;X n are independent normal random variables then a X + 1 1 + ▯▯2 2 a X is n n also normally distributed for any choices of a1;a2;:::;a n Distribution Sampled From N(▯;▯) ▯,
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