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- i)only Let (X1,...,Xn), n ≥ 2, be a random sample from a distribution with discrete probability density fθ,j,whereθ ∈ (0,1), j = 1, 2, fθ,1 is the Poisson distribution with mean θ,andfθ,2 is the binomial distribution with size 1 and probability θ. (i) Show that T = ni=1Xi is not sufficient for (θ,j).The probability that Paul’s train to work is late on any day is 0.15, independently of other days. (i) The number of days on which Paul’s on which Paul’s train to work is late during a 450-days period is denoted by the random carriable Y. Find a value of a such the P(Y>a)≈ 1/6 . In the expansion of (0.15+0.85)^50 , the terms involving 0.15^r and 0.15^(r+1) and denoted by Tr and Tr+1 respectively. (ii) Show that Tr / Tr+1 = 17(r+1)/3(50-r)Let (X, Y ) be a random point in the unit square, [0, 1] × [0, 1], with the uniformprobability. That is, X is a random number between 0 and 1, and Y is a randomnumber between 0 and 1.Your overall task in this problem is to compute P (X < (3/4) given Y < (1/2)X). give answers in fraction please so I can understand what you calculated.
- Let X(n,p) denote binomial random variable with parameter n and p. so that P(X(n,p) > ( greater equal) x) = summation P(X=i) ( summation limit i=x to n ) if W=X(n+1,p) and Y=X(n,p) explay why W>(greater equal "st" Y) See attached for the correct format.Let Y be a continuous random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.
- Let Xi be IID random variables which have the same law as X. Let L(t) = E(e^tX.) Suppose that this is well defined for t ∈ [−1, 1]. Express the moment generating function of the Sum from i=1 to k Xi in terms of k and LLet Y be a discrete random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2Suppose Y is a continuous random variable drawn from the uniform distributionon the interval [3, 4], that is, Y ∼ Uniform([3, 4]). Conditioned on Y = y, a second randomvariable X is drawn from the uniform distribution on the interval [0, y]. What is fX(x), thepdf of X?