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
Q: Any problem for MX (t) at t = 0 ?
1 x/
e
Gamma Distribution. X G(, ). f (x|, ) = x ()
, x, , > 0. E(X) = , V (X) = 2 , MX (t) =
a
(1 t) , t < 1/.
Remark. If X G(k, ) and k cfw_1, 2, ., then P (X x)
Recall that a r.v. X is a map X: S R. X() R.
Def. Given a sample space S, let X1 , ., Xn be n r.v.s from S R. X = (X1 , ., Xn ) is called a discrete
random
all discrete. X is called a cts random vecto
Example 2. Suppose that a bivariate random vector (X, Y ) has a df
f (x, y) = c if x2 + y 2 r2 , where r is a fixed constant.
(1) c =?
(2) F (x, y) = ?
(3) fX (x) = ?
(4) P (X = 2Y ) = ?
(5) P (X > 2Y
4.3. Bivariate Transformation.
Suppose that (X, Y ) is a random vector with joint df fX,Y and (U, V ) = g(X, Y ) = (g1 (X, Y ), g2 (X, Y ).
Two ways to derive fU,V :
(1) cdf or df: If (U, V ) is discr
Pk
It suffices to show that log 1(x > ) 6= i=2 wi ()ti (x).
P
If x > , 0 = ki=2 wi ()ti (x) = w2 ()t2 (x). Thus w2 () = 0 or t2 () = 0.
Pk
If x < , = i=2 wi ()ti (x) = w2 ()t2 (x) = 0. A contradiction
Chapter 4. Multivariate Random Variables
4.1. Recall that a r.v. X is a map X: S R.
X() R.
Def. Given a sample space S, let X1 , ., Xn be n r.v.s from S R, then
X = (X1 , ., Xn ) is called a random ve
Now use Theorem 2 to verify that the limits can be exchanged in Ex. 2:
d
d
Z
xn ex/ /dx =
0
Z
0
d n x/
x e
/dx
d
Let h(x, y) = xn ex/y /y, o = yo /2 (> 0) and write y = yo + , where | < o , y = and yo
E(X(X 1) =
Pn
x=0 x(x 1)
D
)
(Dx)(Nnx
N
(n)
(N D)(N n)
V (X) = E(X 2 ) (E(X)2 = nD
.
N
N (N 1)
Bernoulli Distribution.
A Bernoulli trial is an experiment that there are exactly two outcomes,
called a
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)
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 Independ
Theorem 1. X Y iff fX,Y (x, y) = fX (x)fY (y) for all (x, y), except perhaps for an event A such that P (X, Y )
A) = 0. iff P (f (X, Y ) = fX (X)fY (Y ) = 1 iff P (fX|Y (X|Y ) = fX (X) = 1
Lemma 1. X