1. Math 3680 Lecture #9 The Normal Distribution

2. NORMAL DISTRIBUTION We say X ~ Normal( ,  ) if X has pdf If  = 0 and  = 1, then we say X has a standard normal density:

3. The normal curve – sometimes called the bell curve – is easily the most famous distribution for data. The bell curve pops up in all kinds of applications, such as: SAT scores Attendance at baseball games Brain weights Cash flow of a bank Heights of adult males/females

4. Properties of the Standard Normal Curve: 1.The curve is “bell-shaped” with a maximum at x = 0. 2. It is symmetric about the y-axis 3. The x-axis is a horizontal asymptote, but the curve and the horizontal axis never meet. 4. The points of inflection are at –1 and 1. 5. Half the area lies to the left of 0; half lies to the right. 6. About 68% of the area lies between –1 and 1. 7. About 95% of the area lies between –2 and 2. 8. About 99.7% of the area lies between –3 and 3.

5. PROPERTIES 1. PROOF.

6. PROPERTIES Unfortunately, the cdf cannot be calculated exactly in closed form. To find the cdf of a normal random variable, use numerical integration (e.g. Trapezoid Rule) or the normal table (book, pp. 508/9) after converting to standard units (more on this later).

7. Ex. 0.01618

11. In Excel, the command for the normal( ,  cdf is =NORMDIST(x, , , true) or for the standard normal: =NORMSDIST(x, true) (NORMSDIST(x, false) returns the pdf) On TI calculators, look under the DISTR menu: normalcdf(low, high, ,  ) See p. 224 of the text for more information.

12. PROPERTIES 2. Suppose that X ~ Normal( ,  ). Then

13. PROPERTIES 2. In summary, if X ~ Normal( ,  ), then That is, X has the same distribution as  +  Z. (NOTE: There are many nice consequences of this change of scale.) Definition. We convert X into standard units via

14. Example: Adult men’s heights are normally distributed with  = 70 inches and  = 2.5 inches. What is the probability that a randomly selected man will have a height less than 66 inches? Solution. First convert 66 into standard units:

15. Example: Adult men’s heights are normally distributed with  = 70 inches and  = 2.5 inches. What is the probability that a randomly selected man will have a height greater than 76 inches? Solution. First convert 76 into standard units:

16. Example: Adult men’s heights are normally distributed with  = 70 inches and  = 2.5 inches. What is the probability that a randomly selected man will have a height between 67 and 71 inches?

17. Example: Find the 90th percentile of the distribution of the men’s heights in the previous example. Excel: NORMINV(P, ,  ) or NORMSINV(P) TI: invNorm(P)

18. MOMENTS OF NORMAL DISTRIBUTION 1. Suppose that Z is standard normal. Then (Why?)

19. 2.

20. Therefore, Var( Z ) = E( Z 2 ) - [E( Z )] 2 = 1 - 0 2 = 1 SD( Z ) = 1 Furthermore, if X ~ Normal( ,  ), then E( X ) = E(  +  Z ) =  +  E(Z ) =  Var( X ) = Var(  +  Z ) =  2 E(Z ) =  2 SD( X ) = SD(  +  Z ) = |  SD(Z ) = 