Random Numbers

· Pseudo-random numbers can be generated in MATLAB using the following built-in functions:

1. random – generates most any type of random numbers

2. rand – generates uniform random numbers ranging from 0 to 1

3. randn – generated Gaussian random numbers with mean zero and standard deviation 1

· Use the help command to learn about these functions and try them out

· Learn about the “hist” function for plotting a histogram from a set of data. A histogram is a plot showing how many numbers fall within specific ranges.

· Generate a “bell” curve using the following commands

x=randn(1,1000);

hist(x,20)

xlabel('value')

ylabel('number of occurances')

· Let’s model the throwing of darts for example. The x,y position of each dart relative to the center (0,0) is a random variable. We can generate realizations of these random variables as follows

x=randn(1,1000);

y=randn(1,1000);

figure

plot(x,y, 'rx')

xlabel('x')

ylabel('y')

· Let’s compute the number of darts outside of a 1 unit radius

% radius of each “dart”

r=sqrt(x.^2+y.^2);

I=find( r > 1); % find is a powerful command, use help to learn about it

S=sprintf('The number of darts that were outside a 1 unit radius circle is %d', length(I) );

disp(S)

S=sprintf('The probability that any given dart is outside a 1 unit radius circle is %.2f', length(I)/length(r) );

disp(S)

· What is the probability that after flipping a coin 3 times and getting 3 heads, the next one is heads? Some may think that the probability of getting heads on the 4 try is lower, but in fact this is not true. Given the 3 previous flips are “in the record books” the probability of the next flip is still 50/50. Let’s test this

num_trials = 10000;

heads = rand( 4, num_trials ) > .5;

num_heads_in_4 = sum( heads );

I = find( num_heads_in_4 == 4 );

prob_4_in_4 = length(I)/num_trials;

disp(sprintf('The probability of getting 4 heads in a row is %.2f', prob_4_in_4) )

num_heads_in_first_3=sum( heads(1:3,:) );

I = find(num_heads_in_first_3 == 3);

heads_on_4_after_3 = heads( 4, I );

prob_4_after_3 = sum(heads_on_4_after_3)/length(heads_on_4_after_3);

disp(sprintf('The probability of getting a 4th heads after 3 in a row is %.2f', prob_4_after_3 ) )