4. Let X1,..., Xn be independent, identically distributed random vari- ables with common density 2 log...
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables with common probability function ()p(1-p) m m-a ; x 0,1,. ., m otherwise 0 where m is known and p is unknown (a) Obtain the Sequential Probability Ratio Test of Ho p = po versus HA p P, where pi > po, with significance level 0.01 and power 0.95. Describe the test precisely; (b) For the case where po 3/8,pı = 1/2, m =...
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
Question 1: Suppose that X1, X2,... Xn are independent identically distributed continuous outcome random variables which have a probability density function (pdf) f(z) = π1+ア Calculate (with all working) the pdf of the average of the X,i Comment on the significance of this result to sampling from a random vari- able with the pdf f. This pdf is called a Cauchy density.
2. Let X1, X2,. . , Xn denote independent and identically distributed random variables with variance σ2, which of the following is sufficient to conclude that the estimator T f(Xi, , Xn) of a parameter 6 is consistent (fully justify your answer): (a) Var(T) (b) E(T) (n-1) and Var(T) (c) E(T) 6. (d) E(T) θ and Var(T)-g2. 72 121
Let X1,X2,...,Xn be an independent and identically distributed (i.i.d.) random sample of Beta distribution with parameters α = 2 and β = 1, i.e., with probability density function fX(x) = 2x for x ∈ (0,1). Find the probability density function of the first and last order statistics Y1 and Yn.
2. Suppose that {X1, ..., Xn} are independent and identically distributed random variables from a distribution with p.d.f. See-ox if x > 0 f(x) = 10 if x = 0 Let Y = min <i<n X;. Find the p.d.f. of Y.
Let X1, , X2 ... be a sequence of independent and identically distributed continuous random variables. Say that a peak occurs at time n if Xn-1 < Xn < Xn+1 . Argue that the proportion of time that a peak occurs is, with probability 1, equal to 1/3
The random variables X1, X2, - .. are independent and identically distributed with common pdf 0 х > fx (x;0) (2) ; х<0. This distribution has many applications in engineering, and is known as the Rayleigh distribution. 2 (a) Show that if X has pdf given by (2), then Y = X2/0 is x2, i.e. T (1, 2) i.e. exponential with mean 2, with pdf fr (y;0) - ; y0; (b) Show that the maximum likelihood estimator of 0 is...
Problem 4. Let X1, . . . , Xn be independent with common density f(x) = 2x 1[0 < x < 1]. Set Vn = max(X1, . . . , Xn). . (b) Show that n(1 − Vn) → W in D holds for some random variable W and find the distribution function of W
Let X1,, Xn be independent and identically distributed random variables with unknown mean μ and unknown variance σ2. It is given that the sample variance is an unbiased estimator of ơ2 Suggest why the estimator Xf -S2 might be proposed for estimating 2, justify your answer