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A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point , it moves at random to one of the points , , or , each with probability , independently of its previous moves.
How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$?
If the first step is to the right (which happens with probability $q$), then you must return to the origin; if it is to the left (with probability $1-q$), then you will return to the origin with probability $P_1 = \alpha = q/(1-q)$.
28 Ιουλ 2021 · Considering a discrete random walk in 2D starting from (0, 0) with 1/4 probability of moving in each of the four directions for each step, calculate the probability of returning to (0, 0) after 2n moves. For instance, the probability for the point returing to its origin (0,0) after 2 moves is: 4/42 = 0.25.
Position and displacement. A convenient way to specify the position of an object is with the help of a coordinate system. We choose a fixed point, called the origin and three directed lines, which pass through the origin and are perpendicular to each other.
31 Μαρ 2022 · This article will cover the basics for interpreting motion graphs including different types of graphs, how to read them, and how they relate to each other. Interpreting motion graphs, such as position vs time graphs and velocity vs time graphs, requires knowledge of how to find slope.
24 Απρ 2022 · \(\varepsilon=1\): Equation \ref{final} now becomes \(y^{2}=k^{2}-2 k x\), which is the equation for a parabola (extending along the negative x-axis) with its ‘top’ (in this case, rightmost point) at \((k/2, 0)\) and focal length k/2, so the (single) focus is again located at the origin.
