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- 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?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-Y2dW is normally distributed, dW has mean zero, dW has variance equal to dt. Parameter other than dw is assumed as constant. We have a representation of the geometric Brownian motion as dS/ S = µ dt + σ dW, prove µ dt + σ dW is normally distributed and find its mean and variance.
- Find the moment generating function ME(t) for an exponential random variable with parameter (lambda) = 1. Sketch the graph of ME(t)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.Let X1, . . . , Xn be iid with pdf f(x) = 1 x √ 2πθ2 e − (log(x)−θ1) 2 2θ2 , −∞ < x < ∞, and unknown parameters θ1 and θ2. Find the maximum likelihood estimators for θ1 and θ2, respectively
- A researcher is interested in testing whether annual house hold income in Philadelphia is normal. So she took a sample of 50 house holds and found that skewness (s)= 2.3190 and Kurtosis (k) = 6.7322. Use the Jarque- Bera Test to test , at alpha 0.05, whether income follows normal distribution. -Yes, population is normal because Chi-Square test is higher than critical value. -Yes, population is normal because Chi-Square test is less than critical value. -No, population is not normal because Chi-Square test is higher than critical value. -No, population is not normal because Chi-Square test is less than critical value.Suppose X is a continuous random variable with density f(x) = x/2 , 0 <= x <=2 f(x) = 0 , elsewhere Write an integral expression for the moment generating function M(t).Let X1,...,Xn be iid exponential(θ) random variables. Derive the LRT of H0 : θ = θ0 versus Ha : θ 6= θ0. Determine an approximate critical value for a size-α test using the large sample approximation.
- Consider the time series xt = β1 + β2t + wt, where β1 and β2 are known constants and wt is a white noise process with variance σ2 w. (a) Determine whether xt is stationary. (b) Show that the process yt = xt − xt−1 is stationary. (c) Show that the mean of the moving average vt = 1 2q + 1 q j=−q xt−j is β1 + β2t, and give a simplified expression for the autocovariance function.Suppose that X is uniformly distributed on [0, 4]. Define Y = 2X + 5. Compute the pdf and cdf of Y.The joint pmf of X and Y is f(x, y) = 1/6, 0 ≤ x+y ≤ 2, where x and y are nonnegative integers. Compute Cov(X, Y) and determine the correlation coefficient.