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Math 3680 Lecture #9 The Normal Distribution

Math 3680 Lecture #9 The Normal Distribution. NORMAL DISTRIBUTION We say X ~ Normal( m , s ) if X has pdf If m = 0 and s = 1, then we say X has a standard normal density:.

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Math 3680 Lecture #9 The Normal Distribution

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  1. Math 3680 Lecture #9 The Normal Distribution

  2. NORMAL DISTRIBUTION We say X ~ Normal( m, s ) if X has pdf If m = 0 and s= 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. 0.01618 Ex.

  8. In Excel, the command for the normal(m, s) cdf is =NORMDIST(x, m, s, 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, m, s) See p. 224 of the text for more information.

  9. PROPERTIES 2. Suppose that X ~ Normal( m, s ). Then

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

  11. Example: Adult men’s heights are normally distributed with m = 70 inches and s = 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:

  12. Example: Adult men’s heights are normally distributed with m = 70 inches and s = 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:

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

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

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

  16. 2.

  17. Therefore, Var( Z ) = E( Z2) - [E( Z )]2 = 1 - 02 = 1 SD( Z ) = 1 Furthermore, if X ~ Normal(m, s), then E( X ) = E(m + sZ ) = m + s E(Z ) = m Var( X ) = Var(m + sZ ) = s2 E(Z ) = s2 SD( X ) = SD(m + sZ ) = |s| SD(Z ) = s

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