4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?
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![4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o².
Show that
E(X(X-1)]=µ(µ-1)+o?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b1ac93b-2030-49ee-bbfa-587197c07896%2Ff6d16412-aab6-43d6-a39b-95ac77ab22a8%2F9c5v8ig_processed.png&w=3840&q=75)
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- 3. Consider two random variables X₁ and X2 whose joint pdf is given by for x₁0, x2 > 0, x₁ + x2 < 2, { Find the pdf of Y = X₂ - X₁. f(x1, x₂) = NIT otherwise.1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?2
- 7. a) Suppose that X is a uniform continuous random variable where 03. Let X1, and E- X; are independent. , Xn be iid with exponential distribution with mean 0. Show that (X1+X2)/ E=1 X;5- Suppose X1 and X2 are i.id observations from the pdf fa,p (x) = axa-1e-x" logX1 ,x > 0, a > 0 is an ancillarv statistics. logX2 Show that1. Consider a random sample of size n drawn from a population with mean ux and variance o. Show that: (a) Xn is an unbiased estimator of ux. (E{Xn}= µx or E {(Xn – µx)} = 0) - (b) X = E i where I = E, i, is an unbiased estimator of ux ixi (c) E {(X, – #x)²} = n (d) š? Σ--Χ.)2 is a biased estimator of o. n To prove or show this you will want to rewrite the expression for S2 adding and subtracting µx: E(X; - Hx + Hx - Xn)² n Rearrange this to get to the point where you are taking the expectations of known expressions: expectation from (a) expectation from (c) E{X;} = µx or E{(X; – µx)} = 0 E{(X; – Hx)²} = o%) (e) The sample variance, S? = LisXi-Xn)* is an unbiased estimator of o?. (Use the п-1 same approach as (d))Consider a set of data x1, x2, n n i=1 ..., n taken from a population with mean µ. - Show that (x-μ)² = Σ(x₂ − x)² + n(x − µ)². i=18. Let the random variable X have the pdf 2 x2 fx (x) = exp %3D - V2n 2 Find the mean and the variance of X. Hint: Compute E (X) directly and E (X²) by comparing the integral with the integral representing the variance of a random variable that is N(0,1). i DCO 04 < (X - 5)2 < 38.4).Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON