Example Suppose X1, X2, ..., Xn are iid Bernoulli(0), where 0< 0 < 1. Show that n T = T(X) = X; と ~B(n,e) %3D is a complete statistic. D g(T) @ E [gj(9]
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- Find the moment generating function ME(t) for an exponential random variable with parameter (lambda) = 1. Sketch the graph of ME(t)Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?f X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.
- X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.A simple random sample X1, …, Xn is drawn from a population, and the quantities ln X1, …, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false: a) X1, …, Xn come from a population that is approximately lognormal. b) X1, …, Xn come from a population that is approximately normal. c) ln X1, …, ln Xn come from a population that is approximately lognormal. d) ln X1, …, ln Xn come from a population that is approximately normal.
- 13) Random variables X and Y have joint pdf fXY={4xy, 0≤x≤1, 0≤y≤1fXY={4xy, 0≤x≤1, 0≤y≤1 Find Correlation and CovarianceLet Z~Normal(0,1). Define X1 = Z and X2 = Z^2. Calculate the correlation coefficient ofPearson's correlation coefficient.Hint 1: For Pearson's correlation coefficient, it is possible to use the property of the symmetry of the normal distribution.symmetry of the normal distribution, or equivalently in terms of calculus, the properties of integrals for odd functions.properties of the integrals for odd functions. In this way, it will not be necessary tosolve integrals that have no solution in the form of an easy to work with function. Hint 2: The way I found to calculate both Kendall's and Spearman's coefficients was to calculate the probability P[(X1 - X1).Spearman coefficients, was to calculate the probability P[(X1 - X1′)(X2 - X2′) > 0] and substituting it into theforms equivalent to the definition that come in the book. The one that seems simplest to meis to substitute the X's by their corresponding Z's and use remarkable products to arrive at aprobability in terms of a normal…Let Z~Normal(0,1). Define X1 = Z and X2 = Z^2. Calculate the Spearman's coefficient of concordance for the pair (X1, X2).Hint 1: For Pearson's correlation coefficient, it is possible to use the property of the symmetry of the normal distribution.symmetry of the normal distribution, or equivalently in terms of calculus, the properties of integrals for odd functions.properties of the integrals for odd functions. In this way, it will not be necessary tosolve integrals that have no solution in the form of an easy-to-work function.Hint 2: The way I found to calculate both Kendall's and Spearman's coefficients was to compute theSpearman coefficients was to calculate the probability P[(X1 - X1′)(X2 - X2′) > 0] and substituting it into theforms equivalent to the definition that come in the book. The one that seems simplest to meis to substitute the X's by their corresponding Z's and use remarkable products to arrive at aprobability in terms of a normal distribution.
- Consider a random process which is given by Y(t) = t - Z where Z is a random variable with mean 1.2 and second moment 2.5. The autocovariance of the random process X(t) isLet X1, X2, ... , Xn be independent random variables where Xi ~ Poisson(λi) for i = 1, 2, ... , n. Find the moment generating function of Σi=1n Xi and find the pdf of X1 | Σi=1n Xi = k2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.