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- Find two nonnegative number x and y for which x+3y=30, such that x2y is maximized.Suppose we have the quadratic function f(x)=A(x^2)+2X+C where the random variables A and C have densities fA(x)=(x/2) for 0≤x≤2, and fC(x)=3(x^2) for 0≤x≤1. Assume A and C are independent. Find the probability that f(x) has real rooLet X1, X2, X3 be independent Exp()-distributed random vairables. Determine the pdf of X(1)X(3)
- Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of YQ3: Let X1. ,X, Binomail(1,0). (i) Show that B(1,0) is a member of the exponential class. (ii) Find the minimum variance unbiased estimator (MVUE) of 0.3. Let X be a continuous random variable on the interval [0, 1] having cdf Fx(x) = 3x² − 2x³. Let W = 4 + 3X. (a) What is the range of W? (b) Find a formula for Fw(w). (c) Find a formula for fw(w).
- Q6. (a) Let X; (i = 1, ..,n) be independent random varables E(X;) = µ and var(X;) = of (i = 1, .,n). Let a = E-1 a; X; be an unbiased estimator of u where a; are constants. Find out how the a, are to be chosen if you wish to minimize var(@). (b) The bias and the mean squared error of an estimator Ô of a parameter 6 are defined as Bias (Ô) = b(ê ) = E(@) – 0 and m.s.e (@) = E{(@ – 0)?}, respectively. Show that m.s.e (@) var@) +{b(@ )}°.8. (Same as #4) At a particular gas station, gasoline is stocked in a bulk tank each week. Let random variable X denote the proportion of the tank's capacity that is stocked in a given week, and let y denote the proportion of the tank's capacity that is sold in the same week. The joint pdf of X and Y is given by f(x,y) (3x if 0 ≤ y < x < 1 = 0 otherwise a) Find the marginal probability distributions of X and Y. b) Show that the two random variables are not independent. Using part of your answer to a) is a much shorter solution. Once you've done this with probability, using common sense is VERY straightforward.(ii) Suppose that X₁ and X₂ have joint pdf 2, 10, Compute the joint pdf of random variables Y₁ = X₁ and Y₂ = X2. f(x1, x₂): for 0 < x1 < x₂ < 1 otherwise =
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