Comparing Normal Distributions

typesetting trivia

the expression for the exponential fortion can look really tiny

(1/((Sqrt[2 π])sigma))(E^(-(x-mu)^2/(2 sigma^2)))

... and we get the same thing, without the tiny print, if we use the "Exp[]" function.

(1/((Sqrt[2 π])sigma))Exp[-(x-mu)^2/(2 sigma^2)]

the family of pdfs

a normal pdf where the mean and std dev are also considered "variables" (they are really parameters.)

Clear[f];
f[x_,mu_,sigma_] = (1/((Sqrt[2 π])sigma))Exp[-(x-mu)^2/(2 sigma^2)]

plot a few of these

Plot[{f[x,-2,5],f[x,5,2],f[x,-7,.5],f[x,0,1]},{x,-10,10}]

force a better window

Plot[{f[x,-2,5],f[x,5,2],f[x,-7,.5],f[x,0,1]},{x,-15,15},PlotRange->{-.05,.8}]

Manipulate[Plot[f[x,mu,sigma],{x,-30,30},PlotRange->{-.05,.8}],{mu,-20,20,2},{sigma,.7,5,.5}]

STUDENT WORK

for fixed mu, slide the sigma bar.
for fixed sigma, slide the mu bar.

WHAT EFFECT DOES CHANGING THE PARAMETER HAVE?

areas under these curves

detail for shading an area under the curve for one standard deviation on either side of the mean.
if (x-mu)/sigma = 1, then x=mu+sigma
if (x-mu)/sigma = -1, then x=mu-sigma

CALL THIS DISTRIBUTION X1

Plot[f[x,-2,5],{x,-2-5,-2+5},PlotRange->{-.05,.5},Filling->Axis]

Show[Plot[f[x,-2,5],{x,-2-5,-2+5},PlotRange->{{-15,15},{-.05,.45}},Filling->Axis],Plot[{f[x,-2,5],f[x,5,2],f[x,0,1]},{x,-15,15},PlotRange->{-.05,.45}]]

the shaded area above is the same as the shaded area below

CALL THE DISTRIBUTION THAT IS NOW SHADED X2

Show[Plot[f[x,5,2],{x,5-2,5+2},PlotRange->{{-15,15},{-.05,.45}},Filling->Axis],Plot[{f[x,-2,5],f[x,5,2],f[x,0,1]},{x,-15,15},PlotRange->{-.05,.45}]]

STUDENT WORK

HOW DO WE DESIGNATE THE DISTRIBUTION FOR X1?

HOW DO WE DESIGNATE THE DISTRIBUTION FOR X2?

SHADE THE CORRESPONDING AREA FOR THE OTHER DISTRIBUTION

WHAT DO WE CALL THIS OTHER DISTRIBUTION?

CALCULATE THE SHADED AREA FOR EACH DISTRIBUTION

WRITE THE PROBABILITY STATEMENTS FOR EACH DISTRIBUTION AND RELATE THEM