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

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Chapter1: Starting With Matlab

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4. Question 4: Y is a discrete random variable with PMF S1/3, y=1 p(y) = 2/3, y = 2 (a) Find the MGF for Y. (b) Develop the first two raw moments from MGF and find the variance V(Y). Answers:
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?
3. Let X1, and E- X; are independent. , Xn be iid with exponential distribution with mean 0. Show that (X1+X2)/ E=1 X;
With an American penny, the likelihood of getting H when it is spun on edge is 0.3. If X is the random variable where X(H ) = 1, X(T ) = −1, find the expected value E(X), the variance, Var(X), and express X in its standard form.
The expected value of the sum or difference of two or more functions of a random variable X is the sum or difference of the expected values of the functions. That is, E[g(X)±h(X)] = E[g(X)] ± E[h(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))
If Y = InX has a N (0, 1) distribution then E (X²) is equal to
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
Please answer number2
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?
3. The reliability function (7) of a system, where time is in units of hours, describes the time to failure random variable X, It is given that (t) = 1 (at + 1)² u(t) where is a positive constant, and (1) is the usual unit step function. a) Determine the value of the constant if is known that the MTTF is 100 hours. b) Determine the pdf fx (1) of the random variable X, c) Determine the conditional failure rate ẞ, (t) of the system and plot it as a function of time using the result of part a).

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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].