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- Actuary Tong has to study for two actuarial exams: Exam P and Exam FM. The amount of study time that Actuary Tong will spend on each exam in a day follows a continuous random variable that ranges from o to i hour. The amount of study time that Actuary Tong spends on both exams in a day has a joint density function that is equal to the sum of the study times that Actuary Tong spends on ench exam in a day. Caleulate the probability Actuary Tong spends at least half an hour in a day studying for exactly one of the exams. A 0.45 B 0.50 0.55 D 0.60 0.65 REPYou are a statistical trainee actuary working for an insurance company. You are part of a statistical modelling team that is developing models based on independent non identically distributed random variables. You consider the following model X; ~ Gamma(awi, B), i= {1,..., N} where a, B, and w1,..,WN are positive scalar parameters (the gamma distribution has several common parametrisations, here we define the p.d.f. of a random variable X ~ as f(x) https://mathworld.wolfram.com/GammaFunction.html). Gamma(a, B) Hora-1 exp{-Bx} for x > 0, where r(-) denotes the gamma function, see You have been asked to explore the statistical properties of the maximum likelihood estimator of B assuming that a and wi,... ,WN are known. Assume that a > 2 andE,wi > 1. Let X = X1,..., XN. You decide to perform the following analyses: 1. State the likelihood function L(B; X) = f(X; B). 2. Derive the maximum likelihood estimator for 3, denoted by B(X). 3. Show that the bias of 3(X) is given by B(B) = B/(a Ewi…Describe a bivariate random variable that has cdf H(x,y) = max(0,x) ×max(0,y) forx < 1 and y < 1, and 1 elsewhere
- Let X ~ Binomial(3,0.5) and Y ~ Poisson(2) The two random variables are independent. Calculate the quantity, Var(2X-5Y)Suppose that two continuous random variables X and Y have joint probability density function fxy = 1sxs2,0sy<3 elsewhere Find the strength of the relationship and interpret the findings.Find the moment generating function of the continuous random variable X∼U (a, b).
- Determine the marginal or density function and whether the random variables are independent.If X, Y are normally distributed independent random variables, then show that W = 2X - Y is normally distributed. Do not use moment generating function, only use convolution formula.Please don't copy Q) check if this statement is correct. If it is false, suggest an counterexample. "For random variables X~Bin(n,P/Q), Y~hypergeometric (n,P,Q), V(X)>V(Y)."