Introduction to Probability Theory and Statistics Problem 1. Problem

 ! "# $ % %
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! +*, % &
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0
&1
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#
.
2
X Y ! 15 + 5
3(x + y), 0 ≤ x + y ≤ 1, 0 ≤ x, y < ∞
f (x, y) =
0,
%,
3 X Y 4
5 Y ! X = 1/2
6
.
10 + 10 + 12
/
X Y % 1, 2, 3 4 2
! ⎧
1
⎪
⎨ 16 , x = y
2
f (x, y) = 16
, x<y
⎪
⎩
0, x > y.
5 %
! f1(x) f2(y) X Y 5 Y ! X = 3
5 Y ! X = 3
.
7+5+5
' X E[(X − 1)2] = 10 E[(X − 2)2] = 6
5 X 5 X 5 P
√ X − 7 ≥ 15 2
7
.
8 X ! f (x) =
10 + 15 + 6
e−1−x , −1 < x
%.
0,
5 !! X 5 ! ! "$
5 ! "$
9
.