6. Suppose that X, , ,x, are i.id. randon variables and let X-n Σηί Xi. (a)...
help me 5. Refer to the effect of packaging on sales example in Lecture note 3 and 4. Suppose you are interested in testing whether there is any difference in sales for package designs D3 and D4. Suppose also that σ-10 (a) Write down the null and alternative hypotheses (b) Write down the test statistic. Compute the P- value of the test (c) What is your conclusion (assume α 0.01 level of significance) ? 6. Suppose that, X1,… , xn...
Suppose Xi, X2, . . . , xn are i.id. random variables with Xi ~「α, β). Find the distribution of the sum of the X,'s and the distribution of the average of the X,'s.
suppose Xi, X2, . . . , X, are i.id. random variables with Xi ~ exp(A). Show that Σ-x, ~ T(n, t).
Exercice 5. Let Xi, ,Xn be iid normal randon variables : Xi ~ N(μ, σ2). We denote 4 Tl Show that (İ) ils2 (i.e., that x is independent of 82). (ii) x ~ N(μ, σ2/n). (iii) !뷰 ~ เลี้-1
4. Let Xi,X2, , Xn be n i.id. exponential random variables with parameter λ > Let X(i) < X(2) < < X(n) be their order statistics. Define Yǐ = nX(1) and Ya = (n +1 - k)(Xh) Xk-n) for 1 < k Sn. Find the joint probability density function of y, . . . , h. Are they independent? 15In
Let Xi, X2, X3 be i.id. N(0.1) Suppose Yı = Xi + X2 + X3,Ý, = Xi-X2, у,-X,-X3. Find the joint pdf of Y-(y, Ya, y), using: andom variables. a. The method of variable transformations (Jacobian), b. Multivariate normal distribution properties.
6. Suppose we have i.id. Xi, , Xn ~ N(μ, σ2). In the class, we learned that Σί i m(Xi-X) X2-1. Use this fact and answer the following questions. (a) Consider an estimator σ-c Σηι (Xi-X)2. Find its mean and variance.
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
Problem 1.28. Let Xi, . . . , X, be i.id. Normal(μ, σ2) random variables What is the distribution of (X+-+X,-na)/Vnơ2? How does the central limit theorem work in this case?
Let Xi=I(treatment for ith patient is successful).Then Pr(Xi=1|P=p)=p. Suppose that conditionally P=p, X1,X2,...Xn are independent and X=sum of Xi (x from 1 to n). Suppose P~U(0,1) (uniform distribution), want to find the EX. Could you please show me the steps of this question? Thank you so much!