Question 35. Let X be a random variable defined by the density function if x > B f(x) = xa+1 else with parameters a > 0 and ß > 0. We observe a sample {3, 4, 8}. Which of the following statements are correct? a) The expected value of X exists for all combinations of a and B. b) The expected value does only depend on a, but not on B. c) Given the sample, B can not be larger than 3. 5 d) If we assume B = 2, the estimation of the expected value from the sample mean leads to an estimate of for 3 α. 5 is also the maximum likelihood estimate of a in this case. 3

Exercise 12. Let X, Y, Z ~ N(0, 1) be independent. Determine the distribution, expectation and variance of the following random variables: a) X + Y + Z b) X² + y² + Z²

1. The random variables §, §1, §2,... are independent and identically distributed with 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁, ..., Èn) for n = 1, 2,.... distribution P(§ = 0) = (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X₁ = 3, X₂ = 1|X₁ = 3).

2. Some properties of Expected value and variance of a random variable. a) Assume that X is an arbitrary discrete random variable, and a and b are constant. Using the definitic Show: and E(aX + b) = a · E(X) + b V(aX + b) = a² · V(X ) Stat 3128 Ali Mahzarnia P STAT 3128 Ali Mahzarnia b) Justify the computational formula of Variance of a random variable which is to justify : V(X) = E[(X – µ°] = Ex – µ)P • ptx) = | 2: Here needs justification By Cauchy Schwarz inequality it can be shown that the right hand side is always positive. Analogs expression in Mechanic : Parallel axis theorem Iem = I– md² Moment of Moment of inetria about Inertia of an an axis shifted object about the center of a mass by d from center of mass (a parallel shift) Icm is dispersion around the mean and is like second central moment (variance) I is like second moment if d is mean m is like sum total all the weight of each of the x which all add up to 1 d squared is like squared of mean since we,