5.7.2. Let Y1, 2, ..., Yn be a random sample of size n from a normal pdf having µ = 0. Show that S² = is a consistent estimator for o² = Var(Y). n Y Συ i=1
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- 4 Let f(x) = (1/10)(x-3)^2 for x = 1, 2, 3, 4, 5 %3D Find the standard deviation of X. For the instructor, this was question 11. / 1.8 X 3.1 X 3.4 X 3.5 X 9.4 X 12.411.3.4 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.For a given set of bivariate data, the following result n²100, EX= 5000; EY : 10000; EX2 - 26000; =1010000 and EXY =516000. Ci) Find the predicted value of Y when x = 60. (i) what is the predicted value of the x when Y = 80 (1) Calcutule mie coefficint of Correlation between & and Y. 2ملاعRandom variables X and Y have the joint PMF with a constant k (kxy, х — P(x,y) = { 0, 1,4; y = 1,3 otherwise Find the
- 5. Let X have the pdf (x + 1) -1 < x < 1 f(x) = %3D elsewhere. Find the mean and the variance of X.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.4.5. After fitting the regression model, y= Bo + Bix1 + Brx2+ B3x3+€ on 15 cases, it is found that the mean square error s? =3 and 0.5 0.3 0.2 0.6 0.3 6.0 0.5 0.4 (X'X)-' = 0.2 0.5 0.2 0.7 0.6 0.4 0.7 3.0 Find a. The estimate of V (ß1). b. The estimate of Cov(ß¡, ß3). c. The estimate of Corr(ß1, ß3). d. The estimate of V (ß¡ – B3).
- Suppose a sample Y1, ..., Yn is selected from an exponential distribution with E (Yi) = θ and V (Yi) = θ2. Let θˆ be an estimator of the mean, θ.1. Explain what a parameter is. What is the parameter of interest in this case?2. Name the properties of a good estimator and explain, shortly, what is meant by each. (a)(b) (c)3. Given the properties you discussed in (2) and the parameter of interest identified in (1), what estimator would you propose?4. What would the sampling distribution of this estimator be? Why?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 areSuppose Y₁, Y2,, Yn is an iid sample from a gamma population distribution with shape parameter a = 2 and scale parameter 0, where > 0 is unknown. a. Find a function of the sample mean Y which is an unbiased estimator of 0. Also, find the variance of your unbiased estimator. b. Find two unbiased estimators of 7(0) = 0².