please calculate the CDF
&PDF of the distribution.
please calculate the CDF &PDF of the distribution. (BP) Given X, ~ Uniform[0, 1] for i-1,..,...
That is, the distribution of X has pdf given by θ-11(1 < x 0) and a point mass on {x-1). (b) Let X1, X2,..., Xn be a random sample from the distribution in part (a). Show that the pdf of the maximum order statistic X(n) is given by JX(n) (c) Show that X(n) is a sufficient statistic for θ. Is X(n) complete?
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where θ is an unknown parameter, θ > 0. Let X, be the sample mean and X(n)max(Xi, X2,,Xn). (i) Show that θ¡n-(1 + )Xn ls an unbiased estimator of θ. Find its mean square error and check whether θ¡r, is consistent for θ. (i) Show that nX(n) is a consistent estimator of o (ii) Assume n > 1 and find MSE's of 02n, and compare...
Let X1,... , Xn be a random sample from the Pareto distribution with pdf { f (r0)= 0, where 0>0 is unknown (a) Find a uniformly most powerful (UMP) test of size a for testing Ho 0< 0 versus where 0o>0 is a fixed real number. (Use quantiles of chi-square distributions to express the test) (b) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = log Xi. (Use quantiles...
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0 >0 is unknown (a) Find a 1-a confidence interval for 0 by pivoting the cdf of X(n) = max{X1, ... , Xn}. (b) Show that the confidence interval in (a) can also be obtained using a pivotal quantity
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0...
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0 >0 is unknown (a) Find a 1-a confidence interval for 0 by pivoting the cdf of X(n) = max{X1, ... , Xn}. (b) Show that the confidence interval in (a) can also be obtained using a pivotal quantity
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0...
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xi) and X(n)- (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , xn (iid) Uniform(0,0), E(R)-θ . What happens to E® as n increases? Briefly explain in words why this makes sense intuitively.
216 CHAPTER 5 MULTIPLE RANDOM VARIA 5.10.3. The random variables X1, ... , Xn have the joint PDF (1 0<xi 31; fx1...Xn (21, ... , Xn) = { i= 1,...,n, lo otherwise. Find (a) The joint CDF, Fx1,...,xn(x1, ..., In), (b) P[min(X1, X2, X3) < 3/4).
Let X1,... , Xn be a random sample from the Pareto distribution with pdf Ox (0+1), x > 1, f(z0) where 0>0 is unknown (a) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = T log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval (b) Find a confidence interval for 0 with confidence coefficient 1 - a by pivoting the cdf...
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ uniform(0, 1), find a transformation g) so that X g(U) has the above CDF. (Hint: This entails the quantile function F-().)
2. Suppose that the CDF of X is given by Fur :53 e-3 for x 3....