ii) What is the value of the sample size n, if P[1(X₁ - X)² > 68.3392] = 0.025? i=1
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![b)
Let X₁, X₂, ..., Xn and Y₁, Y₂, ..., Ym be random samples from populations with moment generating
functions Mx,(t) = ³t+t² and My (t) = (₁2)²5, respectively.
ii) What is the value of the sample size n, if P[1(X; - X)² > 68.3392] = 0.025?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc0f7cc9-058c-4420-b4c2-0bfb103b906f%2F316dc07f-36bc-4f12-93ae-2976522a62e0%2Fist0ipv_processed.png&w=3840&q=75)
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- 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.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…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) = 0
- X 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-Y2Suppose 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).Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)
- 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]If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?Let X1, X2, ... , Xn be a random sample, normally distributed with mean μ and variance σ2If σ2 is unknown, find a minimum value for n to guarantee, with probability 0.90, that a 0.95 CI for μ will have length no more than σ/4 explain.
- 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)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 ρxyIf two random variables X and Y are independent with marginal pdfs fx(x)= 2x, 0≤x≤1 and fy(y)= 1, 0≤y≤1 Calculate P(Y/X>2)
