Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Grinstead-Snell-Probability
· Problem 10.3.6
Markov Chains
Mathematics · moment-generating-functions-continuous
Problem
Let
X
X
X
be a continuous random variable whose characteristic function
k
X
(
τ
)
k_X(\tau)
k
X
(
τ
)
is
k
X
(
τ
)
=
e
−
∣
τ
∣
,
−
∞
<
τ
<
+
∞
.
k_X(\tau) = e^{-|\tau|}, \quad -\infty < \tau < +\infty.
k
X
(
τ
)
=
e
−
∣
τ
∣
,
−
∞
<
τ
<
+
∞.
Show directly that the density
f
X
f_X
f
X
of
X
X
X
is
f
X
(
x
)
=
1
π
(
1
+
x
2
)
.
f_X(x) = \frac{1}{\pi(1 + x^2)}.
f
X
(
x
)
=
π
(
1
+
x
2
)
1
.
Related theory
Characteristic Functions And Moment Generating Functions
Browse the problem library