Chapter 3: Normal R.V. http://delfe.tumblr.com/.

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Chapter 3: Normal R.V. http://delfe.tumblr.com/

Normal Distribution: Summary Things to look for: bell curve, Variable: X = the event Parameters: X = the mean 𝜎 𝑋 2 =𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 Density: 𝑓 𝑥 𝑥 = 𝑒 − (𝑥− 𝜇 𝑥 ) 2 (2 𝜎 2 ) 2𝜋 𝜎 2 , 𝑥 𝜖 ℝ 𝔼 𝑋 =X, 𝑉𝑎𝑟 𝑋 = 𝜎 𝑋 2

PDF of Normal Distribution (cont) http://commons.wikimedia.org/wiki/File:Normal_distribution_pdf.svg

PDF of Normal Distribution http://www.oswego.edu/~srp/stats/z.htm

PDF of Normal Distribution (cont) http://commons.wikimedia.org/wiki/File:Normal_distribution_pdf.svg

Using the Z table P(Z > z) P(z1 < Z < z2) area right of z = 1  area left of z P(Z > z) area between z1 and z2 = area left of z1 – area left of z2 P(z1 < Z < z2)

Procedure for doing Normal Calculations Sketch the problem. Write down the probability of interest in terms of the original problem. Convert to standard normal. Convert to CDFs. Use the z-table to write down the values of the CDFs. Calculate the answer.

Example: Normal r.v. (Class) The gestation periods of women are normally distributed with  = 266 days and  = 16 days. Determine the probability that a gestation period is less than 225 days. between 265 and 295 days. more than 276 days. less than 300 days. Among women with a longer than average gestation, what is the probability that they give birth longer than 300 days?

Example: “Backwards” Normal r.v. (Class) The gestation periods of women are normally distributed with  = 266 days and  = 16 days. Find the gestation length for the following situations: longest 6%. shortest 13%. middle 50%.