Question

Generate N binary random variables Xi, i E {1,2,.., N] where X 1 or -1 with equal probability in Matlab using rand or randn. According to central limit theorem, i= 1 should follow normal distribution when N is large. (1) Please plot the theoretical pdf of normal distribution (2) Please estimate the pdf of Vv by generating a lot of instances of Vv (hint: use hist command to get histogram then scaleit) (3) Please plot the theoretical pdf and the estimated pdf in one plot (do they match?) (4) What happens when N increase?

Answer 1

Given that

we have to write the MATLAB CODE:

MATLAB CODE:

function adj = random_graph(n,p,E,distribution,fun,degrees)

adj=zeros(n); % initialize adjacency matrix

switch nargin
case 1 % just the number of nodes, n
p = 0.5; % default probability of attachment
for i=1:n
for j=i+1:n
if rand<=p
adj(i,j)=1; adj(j,i)=1;
end
end
end

case 2 % the number of nodes and the probability of attachment, n, p
for i=1:n
for j=i+1:n
if rand<=p
adj(i,j)=1; adj(j,i)=1;
end
end
end

case 3 % fixed number of nodes and edges, n, E
while numedges(adj) < E
i=ceil(rand*n); j=ceil(rand*n);
if not(i==j) % do not allow self-loops
adj(i,j)=adj(i,j)+1; adj(j,i)=adj(i,j);
end
end

otherwise % pick from a distribution; generate n random numbers from a distribution
Nseq=1; % ensure the while loops start
switch distribution
case 'uniform'
while mod(sum(Nseq),2)==1 % make sure # stubs is even
Nseq = ceil((n-1)*rand(1,n));
end
case 'normal'
while mod(sum(Nseq),2)==1 % make sure # stubs is even
Nseq = ceil((n-1)/10*randn(1,n)+(n-1)/2);
end
case 'binomial'
p=0.5; % default parameter for binomial distribution
while mod(sum(Nseq),2)==1 % make sure # stubs is even
Nseq = ceil(binornd(n-1,p,1,n));
end
case 'exponential'
while mod(sum(Nseq),2)==1 % make sure # stubs is even
Nseq = ceil(exprnd(n-1,1,n));
end
case 'geometric'
while mod(sum(Nseq),2)==1 % make sure # stubs is even
Nseq = ceil(geornd(p,1,n));
end
case 'custom'
% pick a number from a custom pdf function
% generate a random number x between 1 and N-1
% accept it with probability fun(x)
while mod(sum(Nseq),2)==1 % make sure # stubs is even
Nseq = [];
while length(Nseq)<n
x = ceil(rand*(n-1));
if rand <= fun(x)
Nseq = [Nseq x];
end
end
end
case 'sequence'
Nseq = degrees;
end

% connect stubs at random
nodes_left = [1:n];
for i=1:n
node{i} = [1:Nseq(i)];
end

while numel(nodes_left)>0 % edges < sum(Nseq)/2

randi = ceil(rand*length(nodes_left));
nodei = nodes_left(randi); % pick a random node
randj = ceil(rand*length(node{nodei}));
stubj = node{nodei}(randj); % pick a random stub

randii = ceil(rand*length(nodes_left));
nodeii = nodes_left(randii); % pick another random node
randjj = ceil(rand*length(node{nodeii}));
stubjj = node{nodeii}(randjj); % pick a random stub

% connect two nodes, as longs as stubs different
if not(nodei==nodeii & stubj==stubjj)
% add new links
adj(nodei,nodeii) = adj(nodei,nodeii)+1;
adj(nodeii,nodei) = adj(nodei,nodeii);
% remove connected stubs
node{nodei} = setdiff(node{nodei},stubj);
node{nodeii} = setdiff(node{nodeii},stubjj);
end

% remove empty nodes
nodes_left1 = nodes_left;
for i=1:length(nodes_left)
if length(node{nodes_left(i)})==0
nodes_left1 = setdiff(nodes_left1,nodes_left(i));
end
end
nodes_left = nodes_left1;

end

end

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