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Miklós Schweitzer
1993 Miklós Schweitzer
10
probability inequality
probability inequality
Source: miklos schweitzer 1993 q10
October 22, 2021
probability and stats
inequalities
Problem Statement
Let
U
1
,
U
2
,
U
3
U_1 , U_2 , U_3
U
1
,
U
2
,
U
3
be iid random variables on [0,1], which in order of magnitude,
U
1
∗
≤
U
2
∗
≤
U
3
∗
U_1^{\ast} \le U_2^{\ast} \leq U_3 ^ {\ast}
U
1
∗
≤
U
2
∗
≤
U
3
∗
. Let
α
,
p
1
,
p
2
,
p
3
∈
[
0
,
1
]
\alpha, p_1 , p_2 , p_3 \in [0,1]
α
,
p
1
,
p
2
,
p
3
∈
[
0
,
1
]
such that
P
(
U
j
∗
≥
p
j
)
=
α
P(U_j ^ {\ast} \ge p_j)= \alpha
P
(
U
j
∗
≥
p
j
)
=
α
( j = 1,2,3). Prove that
P
(
p
1
+
(
p
2
−
p
1
)
U
3
∗
+
(
p
3
−
p
2
)
U
2
∗
+
(
1
−
p
3
)
U
1
∗
≥
1
2
)
≥
1
−
α
P \left( p_1 + (p_2-p_1) U_3^{\ast} + (p_3- p_2) U_2^{\ast} + (1-p_3) U_1^{\ast} \geq \frac{1}{2} \right) \geq 1-\alpha
P
(
p
1
+
(
p
2
−
p
1
)
U
3
∗
+
(
p
3
−
p
2
)
U
2
∗
+
(
1
−
p
3
)
U
1
∗
≥
2
1
)
≥
1
−
α
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