Consider a linear birth-death process where the individual birth rate is ?=1, the individual death rate is ?= 3, and there is constant immigration into the population according to a Poisson process with rate ?. Please explain and show work!

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(a) State the rate diagram and the generator.?

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(b) Suppose that there are 10 individuals in the population. What is the probability that the population size increases to 11 before it decreases to 9??

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(c) Suppose that ? = 1 and that the population just became extinct. What is the expected time until it becomes extinct again? ? in Greek letter alpha. ?

4. Which is true of the Gaussian function? a. One way of computing the derivative of noisy signal is to smooth with a Gaussian before taking the derivative. Now, because differentiation and convolution are both linear operators we can invert their order and thereby compute the derivative of the smoothed signal by simply convolving the raw signal with the derivative of a Gaussian. b. Gaussian filtering in 2D can be decomposed into two 10 Gaussian filtering operations. c. Convolution of a signal with a Gaussian filter in the spatial domain equates with multiplication of the Fourier transform of the signal with the Fourier transform of the Gaussian (which is a Gaussian also) in k-space. d. A Gaussian filter acts as a low-pass filter. e. All of the above are true.2) GAUSSIAN DISTRIBUTION. a) [5] Write down an expression for a Gaussian distribution. Define each variable. b) [5] Modify the above expression to describe a normalized Gaussian (or simply, "normal") distribution. c) (i) [4] Calculate the value of a normal distribution at the mean.4. Vehicles arrive at a car wash station according to the Poisson process. An average of 4, 5 vehicles per hour arrive at this car wash station. If the washing in the parking area washing a vehicle takes an average of 12.5 minutes, and the washing Times of the vehicles correspond to the exponential distribution? a-) Express the process as a birth-death process, draw a transition rate diagram. Determine the rate of "birth" and the rate of "death" b) what are the chances that the system is empty? (-) what are the chances of up to 2 cars at the car wash station? do what are the chances of an incoming vehicle waiting?