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- 1.1 Consider Y₁, Y2, ‚Yn are i.i.d. random variables with n > 4 and T is the sample mean Y. Also let W = 1 Y¿. Then (A) E(Y₁ +Y₂|W) = E(Y₁+Y₂) (B) E(W|T) = E(Y₁|T) (C) E(Y₁+W|W) = E(Y3 + Y4|W) (D) none of the above.11. Let (yı, Y2, .., Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that –2) In Y; Xản i=1Assume that A, B, and C are subsets of a sample space S with Pr(A) = 0.75, Pr(B) = 0.55, Pr(C) = 0.25. 1. Find Pr(A'), Pr(B'), and Pr(C'). Pr(A') = Pr(B') = %3D Pr(C') = 2. If Pr(A U B) = 0.75., find Pr(A n B). Pr(A n B) = 3. Suppose that we know that B and C are disjoint events. Find Pr[B U C]. Pr[B U C] = 4. Suppose that instead of being disjoint C C B. Find Pr[B U C]. Pr[BU C] =
- (a) Let X1 ~ x² (n, 8) and X2 ~ x² (m, 1). mX1 and its parameters. nX2 (i) Find the distribution of X = (ii) If 1 = 0, name the distribution of X given in 3(a)(i).1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).1. Let y1,.. , Yn be a random sample from a uniform distribution on the interval (0, B). Let B1 2 Li=1 Yi and B, = ) Max(y1,.., Yn) be estimators for B. Find the sampling distribution of B2 Show that B, and B, are unbiased estimators for B. 11.
- 4.2 The probability distribution of the discrete random variable X is f(x) = 3 (*)*** Find the mean of X. 3-x (¹)ª (²³)³—ª, x = 0, 1, 2, 3.11. Let (y1, Y2 ... Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that -2 In Y; xảnb) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?
- 20. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 24.2. Consider a random sample X₁, X2,..., Xn of size n from the Uniform (0, 0) distri- bution, where @ > 0. (i) Show that the p.d.f. q(t; 0) of the sufficient statistic is given by T(X)= max X₁ 1SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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