21 Let x.., X, be ii.d. r.v.s with the Negative Exponential p.d.f., /(x.6)0,ee-(O.) Then , x...
4. Let X have p.d.f. fx(1),-1 < 2. Find the p.d.f. of Y-X2
x, R(x), is defined as the probability that X>x; ie, R(x)-P(X> x). Now, suppose that X has the Negative Exponential p.d.f, Then (ii) Use Theorem 1 in order to determine the MLE of R(x; θ), on the basis of a random sample X1,…, X from the underlying p.d.f. Theorem 1 Let θ-θ(x) be the MLE of θ on the basis of the observed values x1, , xn of the random sample X1, , X, from the pdf f(", θ), θ...
random sample of size n from the p.d.f. 1.8 On the basis of a (x,θ)-θΧθ-1 , 0 < x < 1, θ E Ω = (0,0)derive the MLE of θ
PROB 4
Let Xi and X2 be independent exponential random variables each having parameter 1 i.e. fx(x) = le-21, x > 0, (i = 1,2). Let Y1 = X1 + X2 and Y2 = ex. Find the joint p.d.f of Yi and Y2.
I. Let X be a random sample from an exponential distribution with unknown rate parameter θ and p.d.f (a) Find the probability of X> 2. (b) Find the moment generating function of X, its mean and variance. (c) Show that if X1 and X2 are two independent random variables with exponential distribution with rate parameter θ, then Y = X1 + 2 is a random variable with a gamma distribution and determine its parameters (you can use the moment generating...
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
2) Let X and Y be independent exponential random variables with means E[X] = 0 and EY = 28. 1 1 f(310) = -X/0 e x > 0, f(y|0) = e-4/20 y > 0 0 24 a) Show that the likelihood function can be written as (2 points) L(0) = e-3(x+3) 202 b) Find the MLE ô of 0. (5 points)
Let X1, X2,...,Xn be a random sample from the exponential distribution with rate A Let c > 0 be a fixed and known number. For i 1,2 п, let ..1 -{: : if Xic 1 Y otherwise Suppose that you get to observe Yı, Y2,... , Y,n but you do not get to observe X1, X2,... , X,n п. Find the MLE for X based on this information
PROB5
Let U and V be independent r.v's such that the p.d.f of U is fu(u) = { 2 OSU< 27, otherwise. and the p.d.f'of2 is Seu, v>0, fv (v otherwise. Let X = V2V cos U and Y = 2V sin U. Show that X and Y are independent standard normal variables N(0,1).
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient