1. PATTERN RECOGNITION LAB 2 TA : Nouf Al-Harbi:: nouf200@hotmail.com

2. Lab objective:  Illustrate the uniform distribution of a random variable using Matlab 2

3. Theoretical Concept Part 1 3

4. Suppose a die is rolled. What is the probability that the die will land on 5 ? On 4, on 2 ….? Dice Experiment 4

5.  When a die is rolled  there are 6 possible outcomes  represented by: X = { 1, 2, 3, 4, 5, 6 }.  Each outcome is equally likely to occur  If a die is rolled 1200 times  T hen, each of outcome should occur  1200/6 = 200 times Frequency F(X) Outcome x 2001 2 3 4 5 6 Frequency distribution 5

6. Dice Experiment  What’s the probability for occurring each outcome..?  P(X = 6) = 200/1200 =1/6  P(X=3)=P(X=1)=200/1200=1/6  A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence  If each outcome has the same probability then probability density function is called “uniform distribution” Probability P(X=x) Outcome x 1/61 2 3 4 5 6 probability distribution 6

7. What’s uniform distribution..?  We obtain a uniform density function when the outcomes of an experiment (random process) are equally likely to occur. 7

8. Practical Applying Part 2 8

9. Applying dice experiment by Matlab 1. Generate N random values uniformly distributed in the closed range [1,6]. 2. Find the frequency distribution of each outcome (1-6)  (i.e. how many times each outcome occur?) 3. Find the probability density function p(x) 4. Plot p 9

10. Generate N random values uniformly distributed in the closed range [1,6]. Step 1 10

11. rand function rand(1,N)  Generates N random values uniformly distributed in the open range ]0,1[.  Write the following in Matlab & see the result:  r = rand(1,20)  generates 1-D array of one row and 20 columns  Random values range between 0 and 1  To change the period we can use fix function  11

12. fix function  x = fix( 6 * r ) + 1;  Writing the previous line converts r into random values in the closed period [1,6]  For Dice Experiment, What are the values of vector x represent..? 12

13. Find the frequency distribution of each outcome (1-6) Step 2 13

14. Find the frequency distribution of outcome  we’ll make a counter for each outcome 14 123456 120 191199210300180 Event no.1234567…1200 outcome2112566…2 N

15. Find the probability density function p(x) Step 3 15

16. Find the probability density function p(x) 16  We can easily calculate t he probability  the outcome frequency divided by the no. of events  P=f/N

17. Plot the probability density function p(x) Step 4 17

18. plot p(x) 18  plot function has different forms, depending on input arguments.  If you have two vectors y and x  plot (x,y) produces a graph of y versus x  If you have one vector x  plot(x) produces a graph of columns of x versus their index  To change the axis scale, that is x starts from xmin to xmax and y starts from ymin to ymax use the command: axis([xmin xmax ymin ymax])

19. plot p(x) 19  If we have more than one graph, we can use figure command to create a new figure window  It’s useful to avoid draw the new graph over the previous one  For more information about plot function and its forms type help plot on command window

20. 1.N = 100; 2.r = rand(1,N); 3.x = fix( 6 * r ) + 1; 4.f = zeros(1,6); 5.for i = 1 : N 6.if x(i) == 1 f(1) = f(1) + 1; 7.elseif x(i) == 2 f(2) = f(2) + 1; 8.elseif x(i) == 3 f(3) = f(3) + 1; 9.elseif x(i) == 4 f(4) = f(4) + 1; 10.elseif x(i) == 5 f(5) = f(5) + 1; 11.else f(6) = f(6) + 1; 12.end 13.end 14.F 15.plot(f) 16.axis([0 7 0 1.5]) 17.p = f /N 18.figure, plot(p) 19.axis([0 7 0 0.3]) Full code Try larger values of N: (1000,10000) and notice the graph 20

21. Write a Matlab function to illustrate a uniform distribution of coin experiment. A function should take the number of events N as an argument Exercise 1 21