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1b.STATISTICAL INFERENCE
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- Let X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?Suppose X and Y are random variables with E[XY ] = 6, E[Y ] = 4 and E[X] = 5 Find Cov(X; Y )
- Let X be a random variable with probability mass function P ( X = 1 ) = 1/2 , P ( X = 2 ) = 1/3 , a n d P ( X = 5 ) = 1/6 . Then E[1/x]=?Let X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given by FU (x) := P(U ≤ x) = 0 if x ≤ 0 x if 0 < x < 1 1 if x ≥ 1 1. Compute the cdf of the random variable X1. 2. Compute E(X1) and V ar(X1). 3. Give the method of moments estimators of the unknown parameters a and b. Explain how you construct these estimators!Let X1, ...., Xn be a random sample from a population with θ unknown and given by the density f(x; θ) = ( 1 2θ √2 x e − √2 x θ if x > 0 0 if x ≤ 0 1. Show that E(X) = 2θ 2 and E( √2 X) = θ (Hint: you may use that R ∞ 0 e −z z α−1dz = (α − 1)! for every α ∈ N). 2. Show that the statistic θbn := 1 n Xn i=1 p2 Xi (1) is an unbiased estimator of θ. 3. Give the definition of a consistent estimator. 4. Show that the estimator θbn given in relation (1) is a consistent estimator of θ. 5. Show that the estimator θbn is a minimum variance estimator of θ. (Hint: use the Cramer-Rao inequality given by var(θb) ≥ 1 nE ∂ ln(f(X;θ) ∂θ 2
- Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50Consider 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…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.
- There are two traffic lights on a commuter's route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X1, X2 is a random sample of size n = 2). x1 0 1 2 p(x1) 0.1 0.2 0.7 ? = 1.6, ?2 = 0.44 (a) Determine the pmf of To = X1 + X2. to 0 1 2 3 4 p(to) (b) Calculate ?To. ?To = How does it relate to ?, the population mean? ?To = · ? (c) Calculate ?To2. ?To2 = How does it relate to ?2, the population variance? ?To2 = · ?2Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)Let X1, X2, . . . , Xn be an i.i.d. random sample from a Beta distribution with density: f(x; θ) = Γ(2θ) Γ(θ) 2 x θ−1 (1 − x) θ−1 , 0 < x < 1, θ > 0. Find a sufficient statistic