3. Let X and Y be two independent Poisson random variables with mean 2. Then |A|P[min{X, Y} ≤ 1] = 9e-4 CP[min{X, Y} ≤ 1] = 3e-²(2 − 3e-²) [B] P[min{X, Y} ≤ 1] = (1 - 3e−²)² DP[min{X, Y} ≤ 1] = e−4
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![3. Let X and Y be two independent Poisson random variables with mean 2. Then
|A|P[min{X, Y} ≤ 1] = 9e-4
CP[min{X, Y} ≤ 1] = 3e-²(2 − 3e-²)
[B] P[min{X, Y} ≤ 1] = (1 - 3e−²)²
DP[min{X, Y} ≤ 1] = e−4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf0a5446-7c8f-4e18-a28a-efa1dda355a7%2F68580f32-601b-4005-84fa-a512b5366922%2F2asbldf_processed.png&w=3840&q=75)
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- 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 .Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).Let X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1.(b) Deduce from part (a) that GSn(s) = ens−n.
- A poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?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)Let W1, W2, . . . be an uncorrelated random sequence with mean 0 and variance 1.Define the discrete-time random process {Xn : n ∈ N} := {X1, X2, . . .} by Xn = aXn−1 + Wn(n ∈ N) with a and X0 given. For each of the following two separate cases, find the mean functionmX (n) (n ∈ N) and covariance function CX (m, n) (m, n ∈ N) for the process {Xn : n ∈ N}, anddetermine if it is wide-sense stationary.a. a = 1 and X0 = 0.b. |a| < 1 and X0 is a random variable with mean 0 and variance 1/(1 − a2), uncorrelated withW1, W2, . . . .
- Let X1, X2, X3 and X4 be exponential(1) random variables. Find the joint distribution of X1/(X1+X2+X3+X4) and (X1+X2)/(X1+X2+X3+X4) and (X1+X2+X3)/(X1+X2+X3+X4) using Jacobian method?Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0X 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
- Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxy
