Problem 3 Let X be a continuous random variable with PDF S 4z³ 0 < « <1 fx(x) = otherwise Find P(X ).
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- 5- Suppose X1 and X2 are i.id observations from the pdf fa,p (x) = axa-1e-x" logX1 ,x > 0, a > 0 is an ancillarv statistics. logX2 Show that5. Let (*1, x2, ., Tn) be a random sample from the distribution f (x) = Ax^-1; 0 0. Find the moment estimator of A.Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
- The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…The (continuous) random variable X has pdf f(x)=(x-80)/2500 * e^[-(x-80)/50] for x>80 and 0 otherwise. Find the mode. Give an exact answer.Show process
- Suppose X is a random variable whose pdf is given by f(x) =k(4x-2x^2) 01)EL 466 416 13.) The continuous random variable (RV) X is uniform over [0,1). Given Y = -ln X what is P({0Under the proportional reinsurance scheme of a certain risk S, the risk assumed by the insurer is S^A = aS, and the premium received is ap. Suppose that the initial capital of the insurer to cover said risk is u. Show that the default probability P (aS>ap + u) is an increasing function of a ∈ (0, 1).Suppose that X is a continuous unknown all of whose values are between -5 and 5 and whose PDF, denoted f, is given by f ( x ) = c ( 25 − x^2 ) , − 5 ≤ x ≤ 5 , and where c is a positive normalizing constant. What is the expected value of X^2?3) Assume that a random variable X has moment generating function as follows: e' + e²¹ M(t)=4-2e¹ ∞ Calculate the mean and variance of X. for t3 السؤال 5 If w (x) = Ae^ikx for -L/2Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON