MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
2006 Miklós Schweitzer
11
11
Part of
2006 Miklós Schweitzer
Problems
(1)
simple random walk
Source: miklos schweitzer 2006 q11
9/4/2021
Let
α
\alpha
α
be an irrational number, and denote
F
=
{
(
x
,
y
)
∈
R
2
:
y
≥
α
x
}
F = \{ (x,y) \in R^2 : y \geq \alpha x \}
F
=
{(
x
,
y
)
∈
R
2
:
y
≥
αx
}
as a closed half-plane bounded by a line. Let
P
(
α
,
n
)
=
P
(
X
1
,
.
.
.
,
X
n
∈
F
)
P(\alpha,n) = P(X_1,...,X_n \in F)
P
(
α
,
n
)
=
P
(
X
1
,
...
,
X
n
∈
F
)
, where
X
n
X_n
X
n
is a simple, symmetric random walk that starts at the origin and moves with probability 1/4 in each direction. Prove that
P
(
α
,
n
)
P(\alpha,n)
P
(
α
,
n
)
does not depend on
α
\alpha
α
.
probability and stats