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Miklós Schweitzer
1955 Miklós Schweitzer
3
3
Part of
1955 Miklós Schweitzer
Problems
(1)
Miklós Schweitzer 1955- Problem 3
Source:
9/30/2015
3. Let the density function
f
(
x
)
f(x)
f
(
x
)
of the random variable
ξ
\xi
ξ
bean even function; let further
f
(
x
)
f(x)
f
(
x
)
be monotonically non-increasing for
x
>
0
x>0
x
>
0
. Suppose that
D
2
=
∫
−
∞
∞
x
2
f
(
x
)
d
x
D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx
D
2
=
∫
−
∞
∞
x
2
f
(
x
)
d
x
exists. Prove that for every non negative
λ
\lambda
λ
P
(
∣
ξ
∣
≥
λ
D
)
≤
1
1
+
λ
2
P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}
P
(
∣
ξ
∣
≥
λ
D
)
≤
1
+
λ
2
1
. (P. 7)
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