3. Let random variable X have PMF Py(x). We wish to guess the value of X beforeperforming the actual experiment. If we call our guess x, the expected square of theerror in our guess ise(x)= E(X− x)² ]Show that e() is minimized by x = E[X].

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Answered: 3. Let random variable X have PMF Pr…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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?
4. The standard deviation of X, denoted SD(X), is given by SD(X) = Var(X). Find SD(aX + b) if X has variance of o?.
Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).
3. Let X1, and E- X; are independent. , Xn be iid with exponential distribution with mean 0. Show that (X1+X2)/ E=1 X;
1. 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=1
You are performing a two-tailed test.If α=.002α=.002, find the positive critical value, to three decimal places.zα/2
5. Let X be a positive random variable; i.e., P(X - log(EX) (b) E[log(1/X)] > log[1/EX] (c) E (X³) > (EX)³
Let X ~ F(a, ß). If E(X)= % and E(X²)= a+a² If â = nX? find the method of (n-1)s² | .3 moments estimator of
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?
6. Find the median of an exponential random variable X with parameter 1 = 4. (The median of X is m iff P(X < m) = 0.5.) Does it differ from E(X)? Try to explain.
2. Find E(X) and Var(X) if a. P(X = 0) = 0.2, P(X = 1) = 0.3, and P(X = 2) = 0.5. b. X is the number of successes in 200 Bernoulli trials each with probability of success 0.7.

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3. Let random variable X have PMF Pr (x). We wish to guess the value of X before performing the actual experiment. If we call our guess , the expected square of the error in our guess is e(x) = E(X-x)²] Show that e() is minimized by x = E[X].