1. PATTERN RECOGNITION LAB 3 TA : Nouf Al-Harbi:: nouf200@hotmail.com

2. Lab objective:  Illustrate the normal distribution of a random variable  Draw two normal functions using the Gaussian formula 2

3. Theoretical Concept Part 1 3

4. What’s Normal distribution..?  A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena  Height, weight, people age 4

5. Normal distribution  Certain data, when graphed as a histogram (data on the horizontal axis, amount of data on the vertical axis), creates a bell- shaped curve known as a normal curve, or normal distribution. 5

6. A normal distribution of data means that most of the examples in a set of data are close to the "average," while relatively few examples tend to one extreme or the other. Normal distribution 6

7.  Normal distributions are symmetrical  a single central peak at the mean (average), ( μ ) of the data.  50% of the distribution lies to the left of the mean  50% lies to the right of the mean.  The spread of a normal distribution is controlled by the standard deviation, ( σ ).  The smaller the standard deviation the more concentrated the data. 7

8. Normal Distribution form 8

9. Practical Applying A. Using randn function Part 2 9

10. Normal distribution using randn 1. Generate N random values normally distributed where mu=20, sigma=2 2. Find the frequency distribution of each outcome (i.e. how many times each outcome occur?) 3. Find the probability density function p(x) 10

11. randn  generates random numbers drawn from the normal distribution N(0,1), i.e., with mean ( μ ) = 0 and standard deviation ( σ ) =1  Write the following in Matlab & see the result:  r = randn(1,6)  generates 1-D array of one row and 6columns  To change the normal parameters we can use fix function  11

12. fix  x=fix( σ * r + μ )  x = fix( 2* r + 20);  Writing the previous line convert r into random values normally distributed with  mean ( μ ) = 20  standard deviation ( σ ) =2 12

13. N=1000000; mu = 20; % mean. sigma = 2; % standard deviation. r = randn(1,N); x = fix( sigma * r + mu ); xmax = max(x); f=zeros(1,xmax); for i=1:N j = x(i); f(j) = f(j)+1; end plot(f) p = f / N; figure, plot(p) Full code 13

14. Practical Applying B. Using Gaussian(normal) formula Part 2 14

15. Using Gaussian(normal) formula 15  Instead of generating random values normally distributed (using randn), we can use:  Gaussian formula :

16. Draw 2 normal functions using Gaussian formula 16 1. For first normal function : N(20,2)  Mu1=20 sigma1=2  X=1:40  Apply Gaussian Formula 2. For second normal function :N(15,1.5)  Mu2=15 sigma2=1.5  X=1:40  Apply Gaussian Formula 3. Plot the normal functions

17. mu1 = 20; sigma1 = 2; p1 = zeros(1,40); for x = 1:40 p1(x) = (1/sqrt(2*pi*sigma1))*exp(-(x-mu1)^2/(2*sigma1)); End mu2 = 30; sigma2 = 1.5; p2 = zeros(1,40); for x = 1:40 p2(x) = (1/sqrt(2*pi*sigma2))*exp(-(x-mu2)^2/(2*sigma2)); end x=1:40; plot(x,p1,'r-',x,p2,'b-') Full code 17

18. Write a Matlab function that compute the value of the Gaussian distribution for a given x. That is a function that accept the values of x, mu, and sigma, then return the value of N(mu,sigma) at the value of x. For example: call the function normalfn(x,mu,sigma). Then use it by issuing the command >>normalfn(15, 20,3). %this should return the values of this pdf at x =15. Then, use this function to write another function to draw the normal distribution N(20,2), i.e., with normal distribution with mean = 20 and standard deviation = 2. H.W 2 18