5.3.2. The Students t and Snedecors F Distributions.
If X1 , ., Xn are i.i.d. N (, 2 ), then is estimated by X , with error X , and standardized
X
error T = S/ .
n
Question: Distribution of T ?
Notice that:
T=
X
tn1 =
S/ n
X
/ n
(n1)S 2
2 (n1)
.
Def.
y
2
x
2
F (x,y
F (x,y
4. If (X, Y ) is cts, then F (x, y ) = f (u, v )dudv , f (x, y ) = xy ) , provided that xy )
exists. OW, f can be dened arbitrarily.
AB
4.2. Conditional Distribution and Independence. Recall P (A|B ) = P (B ) . If (X, Y ) is
P
f
(x,y
Formulas: F (x, y ) = (u,v)(x,y) f (x, y ), fY (y ) = P (Y = y ) = P (X < , Y = y ) =
f (6, 0) = 1/36, f (10, 2) = 2/36,
fY (2) = 8/36,
F (5, 1) = (x,y)(5,1) f (x, y ) = f (2, 0) + f (3, 1) + f (4, 0) + f (5, 1) = 1/6.
1/36 if (x, y ) = (2i, 0), i cfw_1,