PDF of Lecture Notes - School of Mathematical Sciences

⎧
⎪⎨ 1 0 < x 2 < 1
f 2 (x 2 ) =
⎪⎩
0 otherwise,
1. DISTRIBUTION THEORY
then f(x 1 , x 2 ) = f 1 (x 1 )f 2 (x 2 ), and the two variables are independent.
4. (X 1 , X 2 ) is uniform on unit disk. Are X 1 , X 2 independent NO.
Proof.
We know:
⎧
1
⎪⎨ 2 √ − √ 1 − x 2
1 − x 2 1 < x 2 < √ 1 − x 2 1
1
f X2 |X 1
(x 2 |x 1 ) =
.
⎪⎩
0 otherwise,
i.e., U(− √ 1 − x 2 1, √ 1 − x 2 1).
On the other hand,
i.e., a semicircular distribution.
⎧
2 √
⎪⎨ 1 − x
2
π
2 −1 < x 2 < 1
f X2 (x 2 ) =
⎪⎩
0 otherwise,
Hence f X2 (x 2 ) ≠ f X2 |X 1
(x 2 |x 1 ), so the variables cannot be independent.
Definition. 1.7.8
The joint CDF of RVS’s X 1 , X 2 , . . . , X r is defined by:
F (x 1 , x 2 , . . . , x r ) = P ({X 1 ≤ x 1 } ∩ {X 2 ≤ x 2 } ∩ . . . ∩ ({X r ≤ x r }).
Remarks:
1. Definition applies to RV’s of any type, i.e., discrete, continuous, “hybrid”.
2. Marginal CDF of X 1 = (X 1 , . . . , X r ) is F X1 (x 1 , . . . , x r ) = F X (x 1 , . . . , x r , ∞, ∞, . . . , ∞).
3. RV’s X 1 , . . . , X r are defined to be independent if
F (x 1 , . . . , x r ) = F X1 (x 1 )F X2 (x 2 ) . . . F Xr (x r ).
4. The definitions above are completely general. However, in practice it is usually
easier to work with the PDF/probability function for any given example.
32