Lecture VI

The order of proof of the normal distribution
function is to start with the standard normal:
1  x2 2
f x  
e
2
◦ First, we need to demonstrate that the distribution
function does integrate to one over the entire
sample space, which is - to . This is typically
accomplished by proving the constant.
◦ Let us start by assuming that

I  e

2
y

2
dy
◦ Squaring this expression yields

I  e
2
2
y

2





dy  e
2
x

2


 
e
y2  x2

2
dy dx
dx

Polar Integration: The notion of polar
integration is basically one of a change in
variables. Specifically, some integrals may be
ill-posed in the traditional Cartesian plane,
but easily solved in a polar space.
◦ By polar space, any point (x,y) can be written in a
trigonometric form:
r x y
2
2
x y 
y  r cos 
x  r sin  
  tan
1
◦ As an example, take
f  x   15  x  1 x 2
2
Some of the results for this function are:
x
F(x)
r
theta
-5.00
-4.00
-3.00
-2.00
-1.00
-0.50
-0.25
0.00
0.25
0.50
1.00
2.00
3.00
4.00
5.00
-2.5000
3.0000
7.5000
11.0000
13.5000
14.3750
14.7188
15.0000
15.2188
15.3750
15.5000
15.0000
13.5000
11.0000
7.5000
5.5902
5.0000
8.0777
11.1803
13.5370
14.3837
14.7209
15.0000
15.2209
15.3831
15.5322
15.1327
13.8293
11.7047
9.0139
0.4636
-0.6435
-1.1903
-1.3909
-1.4969
-1.5360
-1.5538
1.5544
1.5383
1.5064
1.4382
1.3521
1.2220
0.9828
r
2.5
2.0
1.5
1.0
0.5
theta
8
10
12
14
◦ The workhorse in this proof is Greene’s theorem.
We know from univariate calculus that

b
a
d f t 
dt  f b   f a 
dt
◦ In multivariate space, this becomes
 N
S
x
 M y dx dy   Mdx   Ndy


 N
S
x
 M y dx dy   N x dx dy   M y dx dy
S
S
  N dy   M dx


◦ Primary function

2
x
f  x   exp 
2

1.0
0.8
0.6
0.4
0.2
4
2
2
4
r
3.0
2.5
2.0
1.5
1.0
0.5
theta
1.5
2.0
2.5
3.0
M  x, y   
y
f  x, t dt , N  x, y   0

 
x
 f dx dy   N
S
x
i 1  i
S
 f dx dy  
S
 M y dx dy    M dx
4
S
d
b

f r dr d    f r cos , r sin  r dr  d
 a

c

I2   e
2
y

2





dy  e
x
2
2


 
e
y2  x2

2
dx
dy dx
y 2  x 2  r 2 cos 2    r 2 sin 2  


 r 2 cos 2    sin 2  
 r2
dy dx  r dr d
I 
2
2
0


0
re
r
2
2
2
dr d   d  2
0
I  2



2
1 y 2
e
dy  1
2

The expression above is the expression for
the standard normal. A more general form of
the normal distribution function can be
derived by defining a transformation
function.
y  a  bx
ya
x
b
◦ By the change in variable technique, we have
1
f x  
e
2
  y  a 2

 
2

2
b


1
b

Definition 2.2.1. The expected value or mean
of a random variable g(X) noted by E[g(X)] is
  g  x  f  x dx if x is continuous
E g  X   
  xX gx  f  x  if x is discrete
◦ The most general form used in this definition allows
for taking the expectation of the function g(X).
Strictly speaking, the mean of the distribution is
found where g(X)=X, or
Ex   x f x dx


◦ Example
E x   

1
0

xe
x

 x
  xe


 x
  e


0
0
dx

 e
0

x

dx
◦ Theorem 2.2.1. Let X be a random variable and let
a, b, and c be constants. Then for any functions
g1(X) and g2(X) whose expectations exist:
 E[a g1(X) + b g2(X) + c]=
a E[g1(X)] + b E[g2(X)] + c.
 If g1(X)  0 for all X, then E[g1(X)]  0.
 If g1(X)  g2(X) for all X, then E[g1(X)]  E[g2(X)]
 If a  g1(X)  b for all X, then a  E[g1(X)]  b