Example 6: Consider a random process X (t) whose mean is zero and Ryx (t) = A 8 (t). Show that the process X (t) is mean-ergodic.
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Q: Example 10: Show that the random process X (t) = A elat is WSS if and only if E [A] = 0. %3D
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Q: Example 10: Show that the random process X (t) = A elat is WSS if and only if E [A] = 0.
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Q: A5. Give the definitions for strictly- and weakly-stationary random processes.
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Q: Define the random process {Xt} by Xt = et + θ et−1. Show this process is weakly stationary.
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Prove the following property of the compound Poisson process:1. E(xt) = λ t E(Y).1 Suppose that X is a stochastic process with dynamics dXt = µdt +σdWt , where W is a P-Brownian motion. The drift µ and the volatility σ are both constants. Find if there is a measure Q such that the drift of process X under Q is η(∈ R) instead of µ.
- Suppose Xn is an IID Gaussian process, withµX[n]=1, and σ2 X[n]=1Now, another stochastic process Yn = Xn − Xn−1. Please find:(a) The mean µY (n).(b) The variance σ2Y (n).(c) The auto-correlation RY (n, k)Define the random process {Xt} by Xt = et + θ et−1. Show this process is weakly stationary.why is the covariance of a deterministic and a stochastic process 0? This relats to Arithmetic Bronian Motion
- 1. Define a Stochastic process and briefly discuss the meaning of measurability of a stochastic process. 2. Consider the ARMA(1,1) model yt = 0.8yt-1 + et + 0.5et-1 with et ~ WN(0, σe2). Derive the Wold representation of yt. 3. Consider the ARMA(2,1) process Φ(L)Xt = Θ(L)et with Φ(L) = 1 − 1.3L + 0.4L2 , Θ(L) = 1 + 0.4L and et ∼ WN(0, σe2). Obtain its Wold representation. 4.Consider the ARMA(2,2) process given by Xt =0.4Xt−1+0.45Xt−2+et+et−1+0.25et−2 with et ∼WN(0,σe2). 5. Consider the MA(1) process yt = et+1.5et−1 with et ∼ WN(0,σe2). Is the above MA(1) a Wold representation? Why or Why not? If not, obtain a suitable Wold representation.If X1, X2, ... , Xn constitute a random sample of size n from an exponential population, show that X is a consis-tent estimator of the parameter θ.Let {N_1(t)} and {N_2(t)} be two independent Poisson processes with rates λ1=1 and λ2=2, respectively. Find the probability that the second arrival in N_1(t) occurs before the third arrival in N_2(t). Round answer to 4 decimals.
- The manager of a market can hire either Mary or Alice. Mary, who gives you service at an exponential rate 20 customers per hour, can be hired at a rate of $3 per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can hired at a rate of $C per hour. The manager estimates that, on the average, each customer’s time is worth $1 per hour and should be accounted for in the model. Assume customers arrive at a Poisson rate of 10 per hour. a) What is the average cost per hour if Mary is hired? If Alice is hired? b) Find C if the average cost per hour is the same for Mary and Alice.Let (Nt)t>0 be a Poisson process with parameter λ=2, Find the following: (a) E(X3X4)Which of the following processes (Xt)t is weakly stationary? A: Xt = 1:6 + Xt 1 + V tB: Xt = 0:6 Xt-1 +V tC: Xt = 0:8 Xt-1 + V tD: Xt = 0:8 t + 0:6 V t – 1 The term (t) is always assumed to be white noise with variance one