Today’s lesson

1. Today’s lesson • Probability calculations with the standard normal distribution. • Making predictions based on the specification of a normal distribution.

2. Chapter Ten: The Normal Distribution • Definition of Normal Distribution • Using tables of the standard normal distribution. • Solving basic problems with the standard normal distribution. • Central limit theorem for sums and averages

3. Definition of the normal distribution • Familiar bell-shaped curve • Continuous distribution, unimodal, symmetric, rapid fall-off of probability for values far from mean. • Probability density function, φ(z) • Cumulative distribution function, Ф(z)

4. Example Normal Distributions • IQ scores are set to be normal with mean 100 and standard deviation 10 or 15, depending on the form. • ETS examination score results are normal distributed with mean 500 and standard deviation 100.

5. Standard Normal Distribution • I always use Z to denote a standard normal. • E(Z)=0 • var(Z)=1 • Φ(z)=Pr{Z<=z} • Appendix D gives right and two-sided tail areas of standard normal (page 549). • I recommend using cdf tables (Ф(z)).

6. General Normal Distribution • Solve problems about any normal distribution by converting to standard normal. • STANDARDIZE the problem: • standard units=(value-expected value)/standard deviation. • Find probability.

7. Today’s Example Scenario • The winnings W in one play of a game of a game of chance is a normally distributed random variable with expected value -$200 and standard deviation $1000. • Advice: always sketch the distribution you are working with.

8. What is the probability that a gambler will win money in one play of this game of chance? • To win money means that W>0. • Must find Pr{W>0}. • Standardize both sides: • Pr{(W-EW)/σW > (0-(-200))/1000}= Pr{Z>0.2}=1-Ф(0.2)=1-0.5793=0.4207. • Answer is 0.4207. Does it make sense?

9. Prediction Intervals • ASS-U-ME quantity to be predicted Y has a normal distribution with known mean E(Y) and known variance σ2. • 95% prediction interval for Y is the interval between E(Y)-1.960σ and E(Y)+1.960σ. • 99% prediction interval for Y is the interval between E(Y)-2.576σ and E(Y)+2.576σ.

10. Differences between Prediction Intervals and Confidence Intervals • Forms are very similar. • A prediction interval contains an observable future value with specified probability. It is thus easy to know when a prediction interval is incorrect. • A confidence interval contains an unknown parameter with specified “confidence”.

11. What is the 99% prediction interval for the winnings in the next play of the game of chance? • The left end-point is E(W)-2.576σ. • Here, -$200-2.576(1000)=-$200-$2576. • There is a 0.005 probability that the gambler will lose $2776 or more. • The right end-point is E(W)+2.576σ=$2376. • There is a 0.005 probability that the gambler will win $2376 or more.

12. Central Limit Theorem for Sums • ASS-U-ME n independent identically distributed observations (usually called a random sample). • Focus on the sum of the n observations: • Sn=W1+…+Wn

13. Central Limit Theorem for Sums • E(Sn)=nE(W) • The “merry-go-round” principle. • Var(Sn)=nvar(W) • Note that sd(Sn)=n0.5sd(W) • The distribution of Sn is asymptotically normal.

14. What are the expected total winnings after 400 independent plays of this game of chance? • E(S400)=400E(W). • E(S400)=400(-$200)=-$80000. • Notice how quickly the losses mount.

15. Second standard problem • What is the standard deviation of the total winnings after 400 independent plays of this game of chance?

16. Solution • Sd(Sn)=n0.5sd(W) • Sd(S400)=4000.5(1000)=$20,000

17. Third Standard Problem • What is the symmetric 99% prediction interval for S400? • Solution: • Left endpoint is E(S400)-2.576sd(S400) • This is -$80000-2.576($20000)=-$131,520. • That is, there is a 0.005 probability that the gambler will lose $131,520 or more.

18. Third Standard Problem • Right endpoint is E(S400)+2.576sd(S400) • This is -$80000+2.576($20000)=$-28480. • That is, there is a 0.005 probability that the gambler will lose $28,480 or less. • The answer is that the 99% prediction interval is the interval between -$131,520 and -$28,480. • The gambler is very sure to lose a lot of money!

19. Fourth Standard Problem • What is the probability that a gambler will have total winnings that are greater than zero after 400 independent plays of this game of chance?

20. Solution • Standardize • Pr{S400>0}= • Pr{[(S400-E(S400))/sd(S400)] • (0-(-80000))/20000=4. • That is, =Pr{Z>4}=1-Φ(4)=0.00003. • The gambler has almost no chance of winning money after 400 independent plays.

21. Discussion of previous problems • The quantities sought are standard approaches to understanding the level of risk involved in a betting (insurance) strategy. • Realistic problems may require more advanced mathematics or simulation techniques.

22. Central Limit Theorem for Averages • ASS-U-ME n independent identically distributed observations (usually called a random sample). • Focus on the average of the n observations: • Mean=Sn/n=(W1+…+Wn)/n

23. Central Limit Theorem for Averages • E(Mean)=E(Sn)/n=(nE(W))/n=E(W) • The expected value of the mean is the expected value of the random variable that was sampled • Var(Mean)=(nvar(W))/n2=var(W)/n. • Note that sd(mean)=sd(W)/n0.5 • The distribution of Sn is asymptotically normal.

24. Major points covered • Definition of the normal distribution. • Use of the normal distribution tables. • Risk management example problems using the normal distribution. • Central limit theorem for sums. • Central limit theorem for averages.