n 5.6.4. Show that ô² = ΣY? is sufficient for o² if i=1 Y1, Y2,..., Yn is a random sample from a normal pdf with μ = 0.
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- 2.5.1. Show that the random variables X₁ and X₂ with joint pdf ƒ(7₁, 72) = { 1271²2(1-22) 0<+1 <1,0<2₂ <1 0 elsewhere are independent.2.2 Let the variable X have the binomial distribution, that is, p(x) = (") p*(q)"-*, x= 0,1,2, -- ,n If the mean of X is given to be µ = np. Find E[X(X – 1)]. Use results in (i) to find the variance of X. i. ii.Consider the one-way analysis of variance model Xij = µ + a; + Eij, i= 1,.., m, j= 1,..., ni, where ɛij ~ N(0,0²) are independent. Let n= n1 + · ·+ nm, ... ni 1 ni 1 X;. >Xii for i = 1,..., m, and X. = - ΣΣΧ. ni j=1 i=1 j=1 (a) Show that SS(TO) = SS(T) + SS(E), where SS(TO) = E E i=12j=1 (Xij – X..)², m m ni SS(T) Σι (X.-Χ.) and SS(E) -ΣΣ (Χ- X.) . i=1 i=1 j=1
- 9.1.4 X1, X2 and X, are iid continuous uniform random variables. Random variable Y = X, + X2 + X, has expected value E[Y] = 0 and variance of = 4. What is the PDF fx,(x) of X,?3.5.11. Let X, Y, and Z have the joint pdf 3/2 a2 + y? + z2 a? + y? + 22 exp 1+ryz exp 27 2 where -o < x < o, -0 < y <0, obviously dependent, show that X, Y, and Z are pairwise independent and that each pair has a bivariate normal distribution. and -0 < 2 < 0. While X, Y, and Z are3.4 Solve the below problem: Let Y, and Y, have a bivariate normal distribution: 2. 1. 2. Y2-H2 exp 2no,02y1-p2 2(1-p2) {. (ज-र.) + ( ज-र) (ज-पदर) 07 (जब)} Dr= –00 < y, < ∞, -00 < y2 < ∞, Show that the marginal distribution of Y, is normal with mean µz and variance o is given by: 2, 1 f(y2) = 203 ,-00 < Y2 < ∞ 2.
- 5.4.6. Let Ymin be the smallest order statistic in a random sample of size n drawn from the uniform pdf, fy(y; 0) = 1/0, 0 ≤ y ≤0. Find an unbiased estimator for 0 based on Ymin.. Let X1, X2, .,X, be a random sample from a normal population with mean u variance o?. Xn and Så are sample mean and variance respectively. According to this information which one of the following is false? a. o2 O b. Vn (Xn-H)~N(0,1) O c. Var(S) = 204 n-1 Od. E(X;) =7 е.For a correlation of -.64 between X and Y, each 1 _SD change in Zx corresponds to a predicted change of _____________SD in Zy.
- 11.8. Let y be a normal random variable with a constant mean E(y) = 4, constant variance var(y.) = o², and a covariance cov(y, y;) = 0 for t# j. Consider the sample mean j = E %3D A. Show that the variance of the sample mean y is .Let X1, . . . , Xn be iid from Exp(β). Find the distribution of the sample maximum Y = X(n). Is this an exponential distribution? (This question was rejected this morning. I did double check and there is no missing info with this question.)
