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Let

p

be an even non-negative continous function with

\int _{\mathbb{R}} p(x) dx =1

and let

n

be a positive integer. Let

\xi_1,\xi_2,\xi_3 \dots ,\xi_n

be independent identically distributed random variables with density function

p

. Define
\begin{align*} X_{0} & = 0 \\ X_{1} & = X_0+ \xi_1 \\ & \vdotswithin{ = }\notag \\ X_{n} & = X_{n-1} + \xi_n \end{align*}
Prove that the probability that all random variables

X_1,X_2 \dots X_{n-1}

lie between

X_0

and

X_n

\frac{1}{n}

.
Proposed by Fedor Petrov (St.Petersburg State University).

Problem Statement