Suppose that the bacteria are attracted to a food source to their right, so they are...

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Suppose that the bacteria are attracted to a food source to their right, so they are more likely to make a "run" to the right than to the left. We will assume that the probability p of moving right remains constant in time, but is now greater than 1/2. This is a very simple model of chemotaxis, the movement of a cell or organism in the direction of increasing or decreasing concentrations of a chemical signal. (a) (5 pts) Recall that the mean position E(xk) of bacteria is always 0 for the case that p = 1/2. What is the mean position E(x) for the general case, in terms of the time step k, the run length l, and the probability of moving right p? Explain your reasoning. (b) (5 pts) Find the distribution for xk in terms of the time step k, the run length l, and the probability of moving right p i.e., find PN (d) the probability of xk = d after N runs. Suppose that the bacteria are attracted to a food source to their right, so they are more likely to make a "run" to the right than to the left. We will assume that the probability p of moving right remains constant in time, but is now greater than 1/2. This is a very simple model of chemotaxis, the movement of a cell or organism in the direction of increasing or decreasing concentrations of a chemical signal. (a) (5 pts) Recall that the mean position E(xk) of bacteria is always 0 for the case that p = 1/2. What is the mean position E(x) for the general case, in terms of the time step k, the run length l, and the probability of moving right p? Explain your reasoning. (b) (5 pts) Find the distribution for xk in terms of the time step k, the run length l, and the probability of moving right p i.e., find PN (d) the probability of xk = d after N runs.