Let the value of the function G(y), for y> 0, be given by the formula:G(v) = √ (ex².Briefly explain why G(y) = yvariable on the interval [0, 1].Note: Here E denotes expectation value.E[(ex-X³)-1] where X is a uniformly distributed random

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let X be a continuous random variable with PDF fx(x)= = { 10 1/8 1≤x≤9, otherwise. Let Y = h(X) = 1/√X. (a) Find E[X] and Var[X]. (b) Find h(E[X]) and E[h(X)]. Which one is greater? (c) Find E[Y] and Var[Y].
Q3. (a) Define a function g(x) of a random [ 1; if x2X, 0; otherwise variables by g(x)% = Where x, is a real number -o< X, <∞ Show that E [g (X)] = 1-F, (x,)
S:55)
Farmers face uncertainty over their earnings because rain is random. Let a farmer's yield be Y= 0 in the dry outcome and Y = 1 in the wet outcome. However, not all farmers are the same: Probability(Y= 1) = p1= 2/ 3 if type = 1 p0= 1/ 3 if type = 0 All farmers are risk averse, with utility U(Y) = √Y as their utility function. Farmers know their type. (a) Calculate each farmer's expected yield, E(Yi) =pI1 + (1- pi)0, expected utility ,E(Ui) =piU(1) + (1 - pi)U(0), and calculate the utility at E(Yi), U(E(Yi)), for i= 0, 1. Show these all on a graph with utility on the vertical axis and yield on the horizontal axis.
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Consider the Prelec function w(p) = e−1.5(− ln p) . Consider only interior fixed points (so ignore the fixed points w(0) = 0 and w(1) = 1). Let p be the probability corresponding to the point of inflexion and p∗ the fixed point. Which of the following statements is true? (A) p < p∗. (B) p > p∗. (C) p = p∗. (D) There is insufficient information to determine the relative sizes of p, p∗.
A fair coin is tossed three times. The random variable X is defined to be , where h is the number of flips that come up heads. For example, X(HHT) = 2 . What is E[X]?
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8 The failure of a circuit board interrupts work that utilizes a computing system, until a new board is delivered. The delivery time, Y, is uniformly distributed on the interval one to six days. The cost of a board failure and interruption includes the fixed cost co of a new board and a cost that increases proportionally to y². If C is the cost incurred, C = Co + c₂y². (a) Find the probability that the delivery time exceeds three days. (b) In terms of co and c₁, find the expected cost associated with a single failed circuit board.
1. Fix r € N, and suppose that Y₁,..., Yn are independent and identically distributed NegBin(r, ) random variables, each with probability mass function given by ¡v (u | 0) – (1 + r − ¹) (1 - fy y (1 - 0) ³0⁰ for y ¤ N and unknown 0 € (0, 1). Here the number of successes r in the negative binomial is known.
Let X be an Exponential random variable, with the moment generating function (MGF): M(t)= (1-9t)-¹. (a) Derive the MGF above. (b) Derive E(X) using the MGF. (c) Derive V(X) using the MGF. (d) Find the MGF of Y, where Y = 2X-3.
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Let the value of the function G(y), for y> 0, be given by the formula: G(v) = (e²²-2³)-¹dz Briefly explain why G(y) = y variable on the interval [0, y). Note: Here E denotes expectation value. E[(ex-X³)-¹] where X is a uniformly distributed random