Let {Nt}{t > 0} be a Poisson process with rate l=2. Find the probability that we see 3 events in the interval [0,3) and 2 events in the interval (1,6]. Round answer to 5 decimals.
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.In a video game with a time slot of fixed length T, signals are generated according to a Poisson process with rate λ, where T > 1/λ. During the time slot you can push a button only once. You win if at least one signal occurs in the time slot and you push the button at the occurrence of the last signal. Your strategy is to let pass a fixed time s with 0 < s < T and push the button upon the first occurrence of a signal (if any) after time s. What is your probability of winning the game? What value of s maximizes this probability?Show that the random process X(t) =cos(2π fot + θ) Where θ is an random variable uniformly distributed in the range {0, π/2, π, π/3} is a wide sense stationary process .
- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.Let {N(t), t ≥ 0} be a Poisson process with rate λ. Let Sn denote the time of the nth event. Find(a) E[N(5) − N(4)|N(2) = 3]. (b) E[N(5)|S3 = 2]. (c) E[S3].Assume that the variables Y1, Y2,... in a compound Poisson process have Bernoulli distribution with parameter p . Show that the process reduces to the Poisson process of parameter λp.
- Let (Nt)t>0 be a Poisson process with parameter λ=2, Find the following: (a) E(N3N4)Let {Xt : t > 0} be a compound Poisson process of parameter λ. Suppose that each of the summands of this process is constant equal to k ∈ IN . Find the distribution of Xt.For a non-homogeneous Poisson process, the intensity function is given by λ(t) = 5 if t is in (1, 2] or (3, 4]; λ(t) = 3 if t is in (0, 1] or (2, 3]. Find the probability that the number number of observed occurrences in the time period (1.5; 4] is more than 2. Round answer to 4 decimals.
- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;Consider a random process which is given by Y(t) = t - Z where Z is a random variable with mean 1.2 and second moment 2.5. The autocovariance of the random process X(t) isX is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2