The Normal Distribution

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1 The Normal Distribution

2 The Normal Distribution

3 Administrivia o Homework 3 due Friday. Good Milestones: o Problems 1, 2, and 3 done last week o Problems 4 and 5 done this week o Midterm coming up inclass on Wednesday October 18 th o Mix of Multiple Choice and FreeResponse Questions o Allowed one 8.5 x 11in sheet of handwritten notes (no magnifying glasses) o Allowed a calculator that can t connect to internet or store large large data

4 Previously on CSCI 3022 Def: A random variable X is continuous if for some function a and b with a apple b P (a apple X apple b) = Z b a f(x) dx f : R! R and for any numbers The function f has to satisfy f(x) 0 for all x and f(x) dx =1. We call f the probability density function of X. R 1 1 Def: The cumulative distribution function of X is defined such that F (x) =P (X apple x) = Z x 1 f(y) dy

5 The Normal Distribution The normal distribution (aka Gaussian distribution) is probably the most important distribution in probability and statistics. Many populations have distributions wellapproximated by a normal distribution Examples: weight, height, and other physical characteristics, scores on tests, etc

6 The Normal Distribution Def: a continuous random variable has a normal (or Gaussian) distribution with parameters and if its probability density function is given by µ 2 f(x) = 1 p 2 e 1 2( x µ ) 2 If a random variable X is normally distributed we say X N(µ, 2 ) Exploration!

7 The Standard Normal Distribution Def: a normal distribution with parameter values normal distribution µ =0, 2 =1 is called the standard Question: What is the pdf of the standard normal distribution fan =, e ±

8 The Standard Normal Distribution Def: a normal distribution with parameter values normal distribution µ =0, 2 =1 is called the standard A standard normal random variable is typically called Z Recall: The normal distribution does not have a closed form cumulative distribution function We use a special notation to denote the CDF of the standard normal distribution OI ( z ) = PCZ < z )

9 The Standard Normal Distribution The standard normal distribution rarely occurs in real life Instead, it s a reference distribution that allows us to learn about other (nonstandard) normal distributions using a simple transformation Recall: For computing probabilities, having a CDF is just as good (or better) as having a pdf Back in the day, you looked up values of the standard normal CDF in normal tables in the back of probability books

10 The Standard Normal Distribution shaded area = (z) standard normal (z) curve :Z

11 The Standard Normal Distribution Example: What is P (Z apple 1.25)? shaded area = = (1.25) (z) standard normal (z) curve

12 The Standard Normal Distribution Example: What is P (Z 1.25)? Example: What is P (Z apple 1.25)? PCZ Example: How can we compute P ( 0.38 apple Z apple 1.25)?, PTZT 1.25 ) = 1 PCZ? 1.257=1 ) = 1 $4.25 ) = $4.257 (125) PC <1.25 )= ( ) = 01(125) OIC..3D

13 Flip It and Reverse It 2.33 ).gs Example: What is the 99 th percentile $( of N(0, 1)? 2. I Hmm: We have tables that tell us areas. How can we go from an area to a value? FROM TABLE. This is the inverse problem to P (Z apple z) =0.99 # ( 2.32 ) How could you do this with a table? &(2.34 A $( 2.331= ) ( zzz, Slope 0<99 OIC 2, SLOPE 0.99 = = 01 (2.33 AX= SLOPE = ,3263

14 Flip It and Reverse It Example: What is the 99 th percentile of N(0, 1)? Hmm: We have tables that tell us areas. How can we go from an area to a value? This is the inverse problem to P (Z apple z) =0.99 pdf, cdt How could we do this with Python? sapy. stats. norm. ppf ( 0.9 9)

15 Flip It and Reverse It stats.norm has lots of good functions related to normal distributions: pdf, cdf, ppf, etc

16 The Critical Value z Notation: We say is the value of Z under the standard normal distribution that gives a = certain tail area. In particular, it is the z value such that exactly of the area under the curve lies to the RIGHT of z We call z the critical value Z oz St. P ( Z > Zoz ) =. 02 shaded area = P (Z z )= q p, 02

17 The Critical Value z Notation: We say is the value of z under the standard normal distribution that gives a certain tail area. In particular, it is the z value such that exactly of the area under the curve lies to the RIGHT of z Question: What is the relationship between z and the cumulative distribution function? Plz > za ) =x= 1 PCZ? z a) = 1 IOAA ) Question: What is the relationship between Za is THE look z a) and percentiles? At Percentile

18 Nonstandard Normal Distributions Nonstandard normal distributions can be turned into standard normals really really easily Proposition: If X is a normally distributed random variable with mean deviation, then Z is a standard normal distribution if µ and standard Z = 1 Cited : E[z]=fE[ ] Mg =fµpf=o Varlz ) = VAR ( E Mg ) = Vary E) = fzv^r(x) = f. on I

normally distributed random variable with mean deviation, then Z is a

19 Nonstandard Normal Distributions Nonstandard normal distributions can be turned into standard normals really really easily Proposition: If X is a normally distributed random variable with mean deviation, then Z is a standard normal distribution if µ and standard Z = X µ or X = Z + µ X N(µ, 2 ) Z N(0, 1),

20 Brake Lights! (in GrandpaVoice) Example: The time it takes a driver to react to brake lights on a decelerating vehicle is critical to helping to avoid rearend collisions The article FastRise Brake Lamp as a Collision Prevention Device (Ergonomics, 1993: ) suggests that reaction time for an intraffic response to a brake signal from standard brake lights can be modeled as a normal distribution having mean value 1.25 sec and standard deviation 0.46 sec. Question: What is the probability that a reaction time is between 1.0 sec and 1.75 sec? 1.0 (1,01.25) 1.75 > ( ) 6 0, a ( i. og) 10h54's PC 1,0111< Pf. 5422<1.087

21 OK! Let s Go to Work! Get in groups, get out laptop, and open the Lecture 12 InClass Notebook Let s: o Get some more practice computing normal probabilities in Python o Look at the way grading curves are often done o See how we can sample from the standard normal using the BoxMuller method when we don t have stats libraries readily available

Introduction to Statistical Inference and Confidence Intervals

Introduction to Statistical Inference and Confidence Intervals Administrivia o Homework 4 due Friday. o Midterm returned in class on Wednesday C PROBABLY ) Previously on CSCI 3022 Proposition: If X is