4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?

Consider a set of data x1, x2, n n i=1 ..., n taken from a population with mean µ. - Show that (x-μ)² = Σ(x₂ − x)² + n(x − µ)². i=1

16. Suppose the true average growth μ of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is σ², whereas for the second type the variance is 40². Let X₁,..., X be m independent growth observations on the first type [so E(X;) = µ, V(X;) = σ²], and let Y₁,..., Y be n independent growth observations on the second type [E(Y) =μ, V(Y) = 4σ²]. a. Show that the estimator û = 8X + (1 - 8) is unbi- ased for μ (for 0 < 8 <1, the estimator is a weighted average of the two individual sample means). b. For fixed m and n, compute V(u), and then find the value of 8 that minimizes V(i). [Hint: Differentiate V(u) with respect to 8.]

In time-series decomposition, seasonal factors are calculated by Multiple Choice O O O O O SFt (Y) (CMA). SFt= Y/CMAt (CMA+) x (SFt) =Yt. SFt = Yt - CMAt. None of the options are correct.

EL 466 416 13.) The continuous random variable (RV) X is uniform over [0,1). Given Y = -ln X what is P({0

Please answer number2

3. (20%) Let W1 < W2 < W3 < W4 < W5 < W6 be the order statistics of n = 6 independent observations from f(x) = x/2; 0 < x < 2. (a) Find E(W6)