EXAMPLE: Does the normal distribution give a good approximation of the binomial distribution? We'll look at some graphs

with(Statistics):

with(plots):

These parameters are from Example 5.7, pg. 178

n:=2; theta:=0.3; mu:=n*theta; sigma:=sqrt(n*theta*(1-theta));

X:=RandomVariable(Binomial(n,theta));

plot([seq([i,ProbabilityFunction(X,i)],i=0..3)]);

Y:=RandomVariable(Normal(mu,sigma));

plot(PDF(Y,x),x=-1..3);

Put them together:

A:=plot([seq([i,ProbabilityFunction(X,i)],i=0..3)],color=red):

B:=plot(PDF(Y,x),x=-1..3,color=black):

display({A,B});

Now increase n and we see that we get a beautiful approximation!

n:=20; theta:=0.3; mu:=n*theta; sigma:=sqrt(n*theta*(1-theta));

X:=RandomVariable(Binomial(n,theta));

plot([seq([i,ProbabilityFunction(X,i)],i=0..15)]);

Y:=RandomVariable(Normal(mu,sigma));

plot(PDF(Y,x),x=-1..15);

Put them together:

A:=plot([seq([i,ProbabilityFunction(X,i)],i=0..15)],color=red):

B:=plot(PDF(Y,x),x=-1..15,color=black):

display({A,B});