Matlab built-in distributions are all standard distributions, See the summarization below.
Solutions for 1.1.
I‘ve implemented those distributions: illustrate PDF, CDF; Draw random values from them.
(0) Univariate Norm
1 mu= 102; sigma = 15; % mean and standard deviation 2 xmin = 70; xmax = 130; % x limitation for pdf and cdf plot 3 n= 100; % number of points for pdf and cdf plot 4 k= 10000; % number of random draws for histogram 5 6 x= linspace( xmin , xmax , n );%create a set of values ranging from xmin to xmax 7 p= normpdf( x , mu , sigma ); % calculate the pdf 8 c= normcdf( x , mu , sigma ); % calculate the cdf 9 10 figure( 1 ); clf; % create a new figure and clear the contents 11 subplot( 1,3,1 ); 12 plot(x,p); 13 xlabel( 'x' ); ylabel( 'pdf' ); 14 title( 'Probability Density Function' ); 15 16 subplot( 1,3,2 ); 17 plot( x , c ); 18 xlabel( 'x' ); ylabel( 'cdf' ); 19 title( 'Cumulative Density Function' ); 20 21 subplot( 1,3,3 ); 22 %draw k random numbers from a N( mu , sigma ) distribution 23 %here the argument "1" means 24 y = normrnd( mu , sigma , k , 1 ); 25 hist( y , 20 );%20 bars 26 xlabel( 'x' ); ylabel( 'frequency' ); 27 title( 'Histogram of random values' ); 28 29 pause;
(1) Beta
1 alpha= 2; beta = 3; %parameters 2 xmin = 0; xmax = 1; % x limitation for pdf and cdf plot 3 n= 100; % number of points for pdf and cdf plot 4 k= 10000; % number of random draws for histogram 5 6 x= linspace( xmin , xmax , n );%create a set of values ranging from xmin to xmax 7 p= betapdf( x , alpha , beta ); % calculate the pdf 8 c= betacdf( x , alpha , beta); % calculate the cdf 9 10 figure( 1 ); clf; % create a new figure and clear the contents 11 subplot( 1,3,1 ); 12 plot(x,p); 13 xlabel( 'x' ); ylabel( 'pdf' ); 14 title( 'Probability Density Function' ); 15 16 subplot( 1,3,2 ); 17 plot( x , c ); 18 xlabel( 'x' ); ylabel( 'cdf' ); 19 title( 'Cumulative Density Function' ); 20 21 subplot( 1,3,3 ); 22 %draw k random numbers from a N( mu , sigma ) distribution 23 %here the argument "1" means 24 y = betarnd( alpha , beta , k , 1 ); 25 hist( y , 20 );%20 bars 26 xlabel( 'x' ); ylabel( 'frequency' ); 27 title( 'Histogram of random values' ); 28 29 pause;
(1) Exponential
1 gamma=2; % parameters 2 xmin = 1; xmax = 7; % x limitation for pdf and cdf plot 3 n= 100; % number of points for pdf and cdf plot 4 k= 10000; % number of random draws for histogram 5 6 x= linspace( xmin , xmax , n );%create a set of values ranging from xmin to xmax 7 p= exppdf( x , gamma ); % calculate the pdf 8 c= expcdf( x , gamma); % calculate the cdf 9 10 figure( 1 ); clf; % create a new figure and clear the contents 11 subplot( 1,3,1 ); 12 plot(x,p); 13 xlabel( 'x' ); ylabel( 'pdf' ); 14 title( 'Probability Density Function' ); 15 16 subplot( 1,3,2 ); 17 plot( x , c ); 18 xlabel( 'x' ); ylabel( 'cdf' ); 19 title( 'Cumulative Density Function' ); 20 21 subplot( 1,3,3 ); 22 %draw k random numbers from a N( mu , sigma ) distribution 23 %here the argument "1" means 24 y = exprnd( gamma, k , 1 ); 25 hist( y , 20 );%20 bars 26 xlabel( 'x' ); ylabel( 'frequency' ); 27 title( 'Histogram of random values' ); 28 29 pause;
(2) Binomial
1 N= 10; theta = 0.7; % parameters 2 xmin = 0; xmax = 10; % x limitation for pdf and cdf plot 3 n= 100; % number of points for pdf and cdf plot 4 k= 10000; % number of random draws for histogram 5 6 x= linspace( xmin , xmax , n );%create a set of values ranging from xmin to xmax 7 p= binopdf( x , N , theta ); % calculate the pdf 8 c= binocdf( x , N , theta); % calculate the cdf 9 10 figure( 1 ); clf; % create a new figure and clear the contents 11 subplot( 1,3,1 ); 12 plot(x,p); 13 xlabel( 'x' ); ylabel( 'pdf' ); 14 title( 'Probability Density Function' ); 15 16 subplot( 1,3,2 ); 17 plot( x , c ); 18 xlabel( 'x' ); ylabel( 'cdf' ); 19 title( 'Cumulative Density Function' ); 20 21 subplot( 1,3,3 ); 22 %draw k random numbers from a N( mu , sigma ) distribution 23 %here the argument "1" means 24 y = binornd( N , theta , k , 1 ); 25 hist( y , 20 );%20 bars 26 xlabel( 'x' ); ylabel( 'frequency' ); 27 title( 'Histogram of random values' ); 28 29 pause;
(4) Uniform
1 a= 5; b = 10; %parameters 2 xmin = 4; xmax = 11; % x limitation for pdf and cdf plot 3 n= 100; % number of points for pdf and cdf plot 4 k= 10000; % number of random draws for histogram 5 6 x= linspace( xmin , xmax , n );%create a set of values ranging from xmin to xmax 7 p= unifpdf( x , a , b ); % calculate the pdf 8 c= unifcdf( x , a , b ); % calculate the cdf 9 10 figure( 1 ); clf; % create a new figure and clear the contents 11 subplot( 1,3,1 ); 12 plot(x,p); 13 xlabel( 'x' ); ylabel( 'pdf' ); 14 title( 'Probability Density Function' ); 15 16 subplot( 1,3,2 ); 17 plot( x , c ); 18 xlabel( 'x' ); ylabel( 'cdf' ); 19 title( 'Cumulative Density Function' ); 20 21 subplot( 1,3,3 ); 22 %draw k random numbers from a N( mu , sigma ) distribution 23 %here the argument "1" means 24 y = unifrnd( a , b , k , 1 ); 25 hist( y , 20 );%20 bars 26 xlabel( 'x' ); ylabel( 'frequency' ); 27 title( 'Histogram of random values' ); 28 29 pause;
(4) Generate identical replica
1 a= 0; b = 1; 2 n= 10; 3 seed=1; 4 rand('state',seed); randn('state',seed); 5 unifrnd(a,b,n,1) 6 %'state'是对随机发生器的状态进行初始化，并且定义该状态初始值。比如你过一段时间还要使用这个随机数的时候，还能保持当前的随机取值。 7 %比如 8 %randn('state',2013) 9 %a = randn(1) 10 %b = randn(1) 会发现与上一个随机值不一样 11 %如果再定义一次 12 %randn('state',2013) 13 %c = randn(1) 会发现与a的值一样 14 rand('state',seed); randn('state',seed); 15 unifrnd(a,b,n,1)
(5) IQ Probalility
1 normcdf(130,100,15)-normcdf(110,100,15)