iid 2. Let Y1,..., Yn-1 xỉ and consider U = E Y;. n-1 i=1 (a) Show that U~ (b) What is the expectation of U? (c) If o? is a constant, what is the expectation of U? n-1
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- . Find each probability.(a) X ∼ Gaussian (0, 3). Find P[X > 4.5].(b) Y ∼ Gaussian (50, 10). Find P[X < 75]C1. Let X be a continuous random variable with PDF f(x) = ? (2 – a) for –1 1)? (c) Calculate the expectation and variance of X. (d) Let Y (X2 – 1). What is the expectation of Y?(E,-0,) Estimate 0 by minimizing the function where E, is the expected number of j=1 losses in the jth interval and O, is the observed number of losses in the jth interval. (A) 6.0 (B) 6.4 (C) 6.8 (D) 7.2 (E) 7.6 You are given: (i) The probability that an insured will have exactly one claim is 0. (ii) The prior distribution of 0 has probability density function: 3 7(0) = Vo, 0<0<1 2 A randomly chosen insured is observed to have exactly one claim. Determine the posterior probability that 0 is greater than 0.60.
- Find the conditional expectation E(Y|X=1)Find the conditional expectation E(XY = 2)(d) X = the length of a randomly selected rattlesnake Describe the set of possible values for the variable. О0, 1, 2, 3, ... O [0, 0) O (0, 0) О 1, 2, 3, ... State whether the variable is discrete. O discrete O not discrete (e) Z = the sales tax percentage for a randomly selected amazon.com purchase, where n = the total number of possible sales tax percentages for online purchases Describe the set of possible values for the variable. O [0, z,] O z, Z2, Z31 ..., Zn О 1, 2, 3, ..., п O [0, n] State whether the variable is discrete. O discrete O not discrete (f) Y = the pH of a randomly chosen soil sample Describe the set of possible values for the variable. О 1, 2, 3, ..., 14 О (0, 14] O [0, 14] О0, 1, 2, 3, ., 14 State whether the variable is discrete. O discrete O not discreteExercise 3. Let X be a random variable with mean u and variance o2. For a E R, consider the expectation E((X – a)²). a) Write E((X – a)²) in terms of a, µ and o2. b) For which value a is E((X – a)²) minimal? c) For the value a from part (b), what is E((X – a)²)?
