Let X1,...,Xn be a random sample from a distribution with mean μ and variance σ2. Find the value of the following quantities (as a function of μ and σ2). (a) E(X1 −X2 +X3 −X4) (b) Var(X1 − X2 + X3 − X4) (c) E(4X1 −3X2 +2X3 −X4)

Q1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 0

2. The random variable X is the amount of fructose in apples (in ounces) grown at a local orchard and is represented by the following cdf: 2(x +), 1

(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.

Which of the following cannot be a probability?   a: 1   b: 1/2   c: -1   d: 0

The following data show the number of class sessions missed during a semester of SOC221 and the final grade for a sample of 8 students selected at random.                       Number of Sessions Missed                                                   Final Grade                                          (x)                                                                               (y)                                           0                                                                                 96                                           2                                                                                 88                                          12                                                                                68                                           6                                                                                 91                                           8…

Let M₁ (2) and M₂(z) be the m.g.f. of r.v.'s X₁ and X2, respectively. Suppose that M (0) = M₂(0) and M"(0) > M"(0). Which of the following statements is true? The mean of X₁ is greater than the mean of X₂ The variance of X₁ is equal to the variance of X₂. The variance of X₁ is greater than the variance of X₂. The variance of X₁ is less than the variance of X₂