Chapter 4 - Lecture 3 The Normal Distribution

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1 Chapter 4 - Lecture 3 The October 28th, 2009 Chapter 4 - Lecture 3 The

2 Standard Chapter 4 - Lecture 3 The

3 Standard Normal distribution is a statistical unicorn It is the most important distribution in statistics. Most of the random variables out in nature fit naturally to normal distribution. Of course, it is a distribution for continuous random variables Chapter 4 - Lecture 3 The

4 Outline Standard A continuous random variable is said to have a normal distribution with parameters µ and σ 2 if the pdf of X is: f (x; µ, σ 2 ) = We denote this as X N ( µ, σ 2) µ)2 1 e (x 2σ 2, < x < 2πσ 2 Chapter 4 - Lecture 3 The

5 Standard Standard The standard normal curve is the normal curve that has µ = 0 and σ 2 = 1 The standard normal random variable is denoted by Z. The pdf of Z is: f (z; 0, 1) = 1 2π e x 2 2, < z < The cdf of Z is denoted with Φ(z) = P(Z z) Chapter 4 - Lecture 3 The

6 Standard Using Tables One can use the Table A.3 in Appendix A to find the cumulative density distribution of standard normal curve. Find using Table A.3 P(Z < 1.34), P(Z < 0.43) and P(Z > 0.35)? Also by reversing the way we read the table we can find the percentiles of standard normal curve. Find using Table the 10th percentile, 45th percentile and 95th percentile. Chapter 4 - Lecture 3 The

7 Standard Notation We will use the notation z α to denote the point on the measurement axis that has area equal to α to the right of z α. Find z 0.05, z 0.80 and z Chapter 4 - Lecture 3 The

8 Standard Standardizing normal distributions Of course, most of the times in real life the random variables that we are interested do not follow the standard normal curve but a general normal curve. We have available only the Tables for standard normal curve. How will we work with the non-standard normal curve? Chapter 4 - Lecture 3 The

9 Standardizing normal distributions Standard By standardizing the normal distribution as follows: If X N ( µ, σ 2) then Z = X µ N(0, 1) σ Example: I believe that X=the weight of male Penn State Students follow a normal distribution with mean 190 pounds and standard deviation 20. Find P(X < 170), P(X > 180) and P(185 < X < 225). Find the 75th percentile of the weight of male Penn State Students. Chapter 4 - Lecture 3 The

10 If the population distribution of a variable is approximately normal then: Roughly 68% of the values are within 1 SD of the mean Roughly 95% of the values are within 2 SDs of the mean Roughly 99.7% of the values are within 3 SDs of the mean. Chapter 4 - Lecture 3 The

11 Normal moment generating function The moment generating function of a normally distributed random variable X is: Proof? M X (t) = e µt+ σ2 t 2 2 Chapter 4 - Lecture 3 The

12 Relation of normal distribution with Binomial distribution In an experiment we are measuring the IQ of students in Penn State and we find out that IQ N(100, 15 2 ) What is the problem here? How do we solve it? Chapter 4 - Lecture 3 The

13 Approximating Binomial with normal distribution If X B(n, p). Then if np 10, n(1 p) 10 we can approximate X as normal distribution, X N(np, np(1 p)) To find probabilities using the cdf (or to standardize X ) we have to use the continuity correction as follows: ( ) x np P(X x) = Φ np(1 p) Chapter 4 - Lecture 3 The

14 Section 4.3 page , 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68 Chapter 4 - Lecture 3 The