![b) A particle executes an unrestricted random walk on the line starting at the origin. The ith step, Zi, has the following distribution: (i) Find the mean and variance of Z,, and hence find the mean and [41 15 12] variance of X the position of the particle after n steps (ii) Find the probability distribution of X, and state the range of X Explain all the steps in your derivation of the distribution (iii) Calculate the following probabilities in terms of p and q: P(X10 =-2), P(Xs = 1). (iv) Calculate the probabilities uo, u, in terms of p and q, and hence find the probability that the particle returns to the origin for the first time after 14 steps.](http://img.homeworklib.com/questions/7ca760d0-6fef-11ea-8439-5b684e39c53f.png?x-oss-process=image/resize,w_560)
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b) A particle executes an unrestricted random walk on the line starting at the origin. The...
A particle executes a simple unrestricted random walk on the line: a step to the right...
A particle executes a simple unrestricted random walk on the line: a step to the right of length 1 occurs with probability 0.66 and a step to the left of length 1 occurs with probability 0.34. Initially the particle is at the origin. Calculate the probability that the particle will ever return to the origin.
A particle executes a simple unrestricted random walk on the line: a step to the right...
A particle executes a simple unrestricted random walk on the line: a step to the right of length 1 occurs with probability 0.66 and a step to the left of length 1 occurs with probability 0.34. Initially the particle is at the origin. Calculate the probability that the particle will ever return to the origin.
A particle executes a simple unrestricted random walk on the line: a step to the right...
A particle executes a simple unrestricted random walk on the line: a step to the right of length 1 occurs with probability 0 56 and a step to the left of length l occurs with probability 0 44 Initially the particle is at the origin. Select the option that gives an approximate value for the probability that the particle will be less than 6 units from the origin after 100 steps, calculated using an appropriate continuity correction Select one: 0.195...
A particle executes a simple unrestricted random walk on the line: a step to the right...
A particle executes a simple unrestricted random walk on the line: a step to the right of length 1 occurs with probability 0.56 and a step to the let of length 1 occurs with probability 0.44. Initially the particle is at the onigin. Select the option that gives an approximate value for the probability that the particle will be less than 6 units from the origin after 100 steps, calculated using an appropriate continuity correction. Select one: 0.195 0.238 0.280...
A particle executes a simple unrestricted random walk on the line: a step to the right...
A particle executes a simple unrestricted random walk on the line: a step to the right of length 1 occurs with probability 0.56 and a step to the let of length 1 occurs with probability 0.44. Initially the particle is at the onigin. Select the option that gives an approximate value for the probability that the particle will be less than 6 units from the origin after 100 steps, calculated using an appropriate continuity correction. Select one: 0.195 0.238 0.280...
Topic 3 (About CLT and Bayes'Theorem: 10 marks] A particle moves along the line in a random walk....
Topic 3 (About CLT and Bayes'Theorem: 10 marks] A particle moves along the line in a random walk. That is, the particle starts at the origin (position 0) and moves either 2 units to the right or I unit to the left in independent steps. If the particle moves to the right with probability 2/3, its movement at the ih step is a random variable X, with distribution P(x+2)-2/3 P(X,-)=13 The position of the particle after 400 steps is the...
2) Starting from the origin, a heavy particle slides down a frictionless wire in the x,...
2) Starting from the origin, a heavy particle slides down a frictionless wire in the x, y plane. Find the shape of the curve such that the particle reaches the vertical line x b in the shortest time. (Hint This is like the brachistochrone problem, but with a free end point. Go through the derivation of the Euler-Lagrange equations, and be careful to consider what happens to the boundary terms after integrating by parts.)
2. Problem 2.5. Consider a random walk on 10..... which movies left and right with respective...
2. Problem 2.5. Consider a random walk on 10..... which movies left and right with respective probabilities a and p. If the walk is at 0 it transitions to 1 on the next step. If the walk is at k it transitions to k-1 on the next step. This is called random walk with reflecting boundaries. Assume that k 3, =1/4, p = 3/4, and the initial distribution is uniform. For the following, use technology if needed. (a) (10.1.X2 }...
5. Let X be a discrete random variable. The following table shows its possible values r...
5. Let X be a discrete random variable. The following table shows its possible values r and the associated probabilities P(X -f(x) 013 (a) Verify that f(x) is a probability mass function (b) Calculate P(X < 1), P(X < 1), and P(X < 0.5 or X > 2). (c) Find the cumulative distribution function of X ompute the mean and the variance of
5. Let X be a discrete random variable. The following table shows its possible values associated...
5. Let X be a discrete random variable. The following table shows its possible values associated probabilities P(X)( and the f(x) 2/8 3/8 2/8 1/8 (a) Verify that f(x) is a probability mass function. (b) Calculate P(X < 1), P(X 1), and P(X < 0.5 or X >2) (c) Find the cumulative distribution function of X. (d) Compute the mean and the variance of X.
A particle executes a simple unrestricted random walk on the line: a step to the right of length 1 occurs with probability 0 56 and a step to the left of length l occurs with probability 0 44 Initially the particle is at the origin. Select the option that gives an approximate value for the probability that the particle will be less than 6 units from the origin after 100 steps, calculated using an appropriate continuity correction Select one: 0.195...
A particle executes a simple unrestricted random walk on the line: a step to the right of length 1 occurs with probability 0.56 and a step to the let of length 1 occurs with probability 0.44. Initially the particle is at the onigin. Select the option that gives an approximate value for the probability that the particle will be less than 6 units from the origin after 100 steps, calculated using an appropriate continuity correction. Select one: 0.195 0.238 0.280...
A particle executes a simple unrestricted random walk on the line: a step to the right of length 1 occurs with probability 0.56 and a step to the let of length 1 occurs with probability 0.44. Initially the particle is at the onigin. Select the option that gives an approximate value for the probability that the particle will be less than 6 units from the origin after 100 steps, calculated using an appropriate continuity correction. Select one: 0.195 0.238 0.280...
Topic 3 (About CLT and Bayes'Theorem: 10 marks] A particle moves along the line in a random walk. That is, the particle starts at the origin (position 0) and moves either 2 units to the right or I unit to the left in independent steps. If the particle moves to the right with probability 2/3, its movement at the ih step is a random variable X, with distribution P(x+2)-2/3 P(X,-)=13 The position of the particle after 400 steps is the...
2) Starting from the origin, a heavy particle slides down a frictionless wire in the x, y plane. Find the shape of the curve such that the particle reaches the vertical line x b in the shortest time. (Hint This is like the brachistochrone problem, but with a free end point. Go through the derivation of the Euler-Lagrange equations, and be careful to consider what happens to the boundary terms after integrating by parts.)
2. Problem 2.5. Consider a random walk on 10..... which movies left and right with respective probabilities a and p. If the walk is at 0 it transitions to 1 on the next step. If the walk is at k it transitions to k-1 on the next step. This is called random walk with reflecting boundaries. Assume that k 3, =1/4, p = 3/4, and the initial distribution is uniform. For the following, use technology if needed. (a) (10.1.X2 }...
5. Let X be a discrete random variable. The following table shows its possible values r and the associated probabilities P(X -f(x) 013 (a) Verify that f(x) is a probability mass function (b) Calculate P(X < 1), P(X < 1), and P(X < 0.5 or X > 2). (c) Find the cumulative distribution function of X ompute the mean and the variance of
5. Let X be a discrete random variable. The following table shows its possible values associated probabilities P(X)( and the f(x) 2/8 3/8 2/8 1/8 (a) Verify that f(x) is a probability mass function. (b) Calculate P(X < 1), P(X 1), and P(X < 0.5 or X >2) (c) Find the cumulative distribution function of X. (d) Compute the mean and the variance of X.
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