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- According to the Maxwell-Boltzmann law of theoretical physics, the pdf of V, the velocity of a gas molecule is given by kv² e-av² v > 0 f,(0) = elsewhere where a is a constant depending on its mass and the absolute temperature and k is an appropriate constant. Show that the kinetic energy Y = mV² is a random variable having Gamma distribution.5. Let Z be a standard normal random variable. Obtain moment generating function (mgf) of Z.Find the density of random variable Y = Z2 and verify that it is a pdf.Assume you fit a logistic regression for binary Y [i.e., replace EY in linear regression by log(EY/(1- EY))=log(odds)]. Instructor claims: beta for the main effect is log(OR), and beta for interaction effect is log(ROR), where ROR denotes the ratio of the odds ratio. Explain your agreement or disagreement.
- Two real-valued RVs, X and Y, have joint PDF 1 p(x1, x2) = exp 2TV1- 2(1 - r?) where -1Consider independent observations (rı, y1), ... (rns yn), from the model Y, Bin(ri, p) for i = 1, ... , n, where the r, are fixed constants. Using likelihood L(p) and log-likelihood I(p) as appropriate, compute the following items. N 1. Derive the maximum likelihood estimate p. 2. Write the second derivative of log-likelihood /(p). == 3. Give an expression of the approximated asymptotic standard error of p by plugging in the estimate p. To this end, estimate the Fisher Information Matrix by : and then s. e. ap² 4. Consider data (15,11), (20,14), (15,9), (10,7), (25,17), (15,12), (10,8). Using your formulæ, compute and write numerical estimates p, s. e. (p) and give a 95% confidence interval for p using the normal approximation. e. (p) = √v¹¹ p=p Note: To answer this question you will work by hand. Do not write in the textbox but upload a single page pdf image of your workings and results. For theoretical computations (a) to (c) you are expected to show your equations and developments…Please don't provide handwrittin solution...SEE MORE QUESTIONS