Consider a birth-and-death process with infinitesimal parameters Ae-5 for k> 0 and μk-15 for k 1....
Problem 10: 10 points Consider a birth-and-death process with infinitesimal parameters, λ,-5 for k20 and A4k-15 for k21. 1. Derive the limiting distribution of X(t), as t → oo. 2. Find the limiting expectation of X(t), as t → oo. 3. Find the limiting variance of X(t)
A birth and death process with parameters λn = 0 for all n ≥ 0 and µn = µ for all n ≥ 1 is called a pure death process. Find Pij (t) for this process.
1. Let {Xt;t >0} be a pure birth process with rate 1x > 0, for x € S = {0,1,2,...}. (a) Write the backward equations (KBE) and use it to solve for Prz(t). (b) Use the result to part (a) to show that the waiting time in state x, say Wx, is exponentially distributed (c) Suppose 1x = 1 is constant for all x E S. Prove by induction that Px-kx(t) = (at) ke Af/k! for k = 0,..., and...
Problem 7 (15 points). Let X be random variable with the binomial distribution with parameters n and 0 <p<1. (1) Show that **- 1 = 2* for any 1 Sxsn. (2) Show that when 0 < x < (n + 1)p, P(X = x) is an increasing function x and for (n + 1)p <x Sn, P(X = x) is a decreasing function x. (3) A certain basketball player makes a foul shot with probability 0.80. Determine for whal value...
5. For the Weibull distribution with parameters a and X, recall that for t> 0 the density function and distribution function are, respectively, f(t) = 410-1-(At) F(t) = 1 -e-(1)" Suppose that T has the Weibull distribution with parameters a = 1/2 and X = 9. (a) (4 points) Compute exactly P(1 <1 < 1.017 > 1). Show your work. Write your answer to 6 decimal places. (b) (4 points) Compute an approximation of P(1 <T < 1.01 T >...
Measurement of a blood test is a random variable X with cumulative distribution function given by 0, 1, r >2 a. Find fx(x), the probability density function b. Graph fx(x) c. Find the mean and the variance of X d. Find the median of X
Problem 1 Assume that the circuit in Fig. 1 has reached steady state byt-0-. The switch is opened at t 0 1. Determine i(0+) and v(0+) dt 3. Find i(oo) and v(oo) 4. Write down i(t) for t>0 0.01F - 142 15 V 4? 1 H Figure 1
Problem 3 A discrete random variable Y takes values {k= 0, 1, 2, ...,} such that PLY Z k} = ()* for k 20. 1. Derive P[Y = k) for any k > 0. 2. Evaluate expectation, E[Y] = 3. Given E[Y(Y - 1)] = 15 , find variance of Y, Var[Y] =
2. The Pareto random variable with parameters a > 0 and B >0 has probability density function (a) Verify that fx is a density function. (b) Find P[X> 3a) (c) Find the mean and variance of X. What restriction do you have on 3 in computing the mean and variance (a different restriction for each)? (d) Use the probability transform to simulate 1000 Pareto random variables with α-1 and β-5 and find their sample mean and variance. Compare this to...
6. Consider the equation ſi 0 -17 |x=b, x>0. [1 1 1 Find all the basic feasible solutions x for these values of b: []: [] [] 67 (-2): [ - ] [-] Draw a picture of the set of all b if x is feasible.