Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
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Chapter 12.3, Problem 17E
a.
To determine
Calculate the value of
b.
To determine
Prove that
c.
To determine
Prove that
d.
To determine
Prove that
e.
To determine
Provide another example that illustrates the same phenomenon.
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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…
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Chapter 12 Solutions
Mathematical Statistics with Applications
Ch. 12.2 - Suppose that you wish to compare the means for two...Ch. 12.2 - Refer to Exercise 12.1. Suppose that you allocate...Ch. 12.2 - Suppose, as in Exercise 12.1, that two populations...Ch. 12.2 - Refer to Exercise 12.3. How many observations are...Ch. 12.2 - Suppose that we wish to study the effect of the...Ch. 12.2 - Refer to Exercise 12.5. Consider two methods for...Ch. 12.2 - Refer to Exercise 12.5. Why might it be advisable...Ch. 12.2 - The standard error of the estimator 1 in a simple...Ch. 12.3 - Consider the data analyzed in Examples 12.2 and...Ch. 12.3 - Two computers often are compared by running a...
Ch. 12.3 - When Y1i, for i = 1, 2,, n, and Y2i, for i = 1,...Ch. 12.3 - Prob. 12ECh. 12.3 - Prob. 13ECh. 12.3 - Prob. 14ECh. 12.3 - A plant manager, in deciding whether to purchase a...Ch. 12.3 - Muck is the rich, highly organic type of soil that...Ch. 12.3 - Prob. 17ECh. 12.4 - Prob. 18ECh. 12.4 - Prob. 19ECh. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Prob. 25ECh. 12.4 - Prob. 26ECh. 12.4 - Complete the assignment of treatments for the...Ch. 12 - Prob. 28SECh. 12 - Prob. 29SECh. 12 - Prob. 30SECh. 12 - Prob. 31SECh. 12 - Prob. 32SECh. 12 - Prob. 33SECh. 12 - Prob. 34SECh. 12 - The earths temperature affects seed germination,...Ch. 12 - An experiment was conducted to compare mean...Ch. 12 - Prob. 37SE
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- Let x and y be random variable such that the mean and variance of X are 2 and 4. respectively, while the mean and variance of y are 6 and k, respectively. A sample of size 4 is taken from the x-distribution and a sample of size 9 is taken from the y-distribution .If p[(x-y)>8]=0.0228,then what is the value of the constant k?arrow_forwardLet X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Let Y1=(X1/r1)/(X2/r2) and Y2=X2. (a) Find the joint pdf of Y1 and Y2.arrow_forwardLet X, Y & Z be independent uniform random variables on the interval [ 0,1 ] . let W= min {X,Y,Z} What is the expected value of(1- w/a)2arrow_forward
- Let X1, · · · , Xn be drawn independently from a distribution with density 1. f(x|θ) = 2/β, θ − 0.25β < x < θ + 0.25β, where β is a positive constant. 2. f(x|θ) = 2x/θ 2 , 0 < x < θ, θ > 0. 3. f(x|θ) = e θ−x for x > θ. For each of the above three cases, construct a 95% confidence interval for θ.arrow_forwardLet X1,…,Xn be i.i.d. Exp(λ) random variables, where λ is unknown. What is the distribution of min1≤i≤n(Xi)? Enter the pdf fmin(x) of mini(Xi) in terms of x. fmin(x) and give an unbiased estimator θ^ for 1/λ.arrow_forwardLet Y1 and Y2 be independent random variables that are both uniformly distributed on the interval (0, 1). Find P( Y1 < 2Y2 | Y1 < 3Y2).arrow_forward
- Let X₁,X₂,...,Xₙ denote a random sample from a distribution that is N(0,θ), where the variance θ is an unknown positive number. Show that there exists a uniformly most powerful test of size α for testing the simple hypothesis H₀ : θ = θ', where θ' is a fixed positive number.arrow_forwardo test H0: μ=40 versus H1: μ<40, a random sample of size n=25 is obtained from a population that is known to be normally distributed. If x=37.1 and s=14.8, compute the test statistic of t0 and If the researcher decides to test this hypothesis at the α=0.1 level of significance, determine the critical value.arrow_forwardLet Y be a discrete random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2arrow_forward
- 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]arrow_forwardDetermine the value c so that f(x) can serve as a probbaility distribution of rhe discrete random variable X: f(x)=c(x+89)/100 For x=0,1,2arrow_forwardLet X1,X2, . . . , Xn be a random sample from a N(μ0, σ2 = θ) distribution,where 0 < θ < ∞and μ0 is known. Show that the likelihood ratio test of H0 : θ = θ0versus H1 : θ = θ0 can be based upon the statistic W =ni=1(Xi − μ0)2/θ0.Determine the null distribution of W and give, explicitly, the rejection rule for alevel α test.arrow_forward
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