3. Four mutual friends are avid texters: at any time, each friend has a probability Xh+o(h)...

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3. Four mutual friends are avid texters: at any time, each friend has a probability Xh+o(h) of sending a text message in the next time interval of length h. Whenever any friend sends a message, it is addressed to one of the other three, with the recipient chosen at random with equal probability. Unknown to the four friends, their messaging platform is susceptible to a virus that spreads whenever an infected device sends a message to an uninfected device. Once infected, a device remains infected for an exponential time with mean 1/μ before security features remove the virus. Let X(t) be the number of infected devices at time t. The four friends do not have a group chat channel and only ever send single-recipient mes- sages. Messages sent to infected devices, or from one uninfected device to another, do not affect the virus. The friends have disabled app updates, so devices can be reinfected even after the virus has been removed. (a) Find the Q-matrix for the continuous time Markov chain (X(t): t ≥ 0). [5] the joint (b) Let J = inf {t > 0: X(t) ‡ X(0)} be the time of the first change in the number of infected devices. Suppose that initially 2 devices are infected. Calculat distribution of J and X(J). Give the average of time J, and the average number of devices infected at time J. [4] 3. Four mutual friends are avid texters: at any time, each friend has a probability Xh+o(h) of sending a text message in the next time interval of length h. Whenever any friend sends a message, it is addressed to one of the other three, with the recipient chosen at random with equal probability. Unknown to the four friends, their messaging platform is susceptible to a virus that spreads whenever an infected device sends a message to an uninfected device. Once infected, a device remains infected for an exponential time with mean 1/μ before security features remove the virus. Let X(t) be the number of infected devices at time t. The four friends do not have a group chat channel and only ever send single-recipient mes- sages. Messages sent to infected devices, or from one uninfected device to another, do not affect the virus. The friends have disabled app updates, so devices can be reinfected even after the virus has been removed. (a) Find the Q-matrix for the continuous time Markov chain (X(t): t ≥ 0). [5] the joint (b) Let J = inf {t > 0: X(t) ‡ X(0)} be the time of the first change in the number of infected devices. Suppose that initially 2 devices are infected. Calculat distribution of J and X(J). Give the average of time J, and the average number of devices infected at time J. [4]

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a To find the Qmatrix for the continuoustime Markov chain Xt t 0 we need to determine the transition rates between different states of the system Lets ... View the full answer