Let X₁,, Xn be a random sample from a population with probability mass function (pmf) p(x) = 0 (1 - 0)¹-, x=0 or 1, 0≤ 0 ≤ 1. 1. ) Find E(X). 2. Find the method of moment (MOM) estimator of 0. 2. ) Calculate a point estimate using this estimator when x₁ = 1, 2 = 1, x3 = 0, 4 = 1, x5 = 0.

N= 150 observations were collected on a time series that was identified as a AR(2) time series. The following statistics were computed from the data. Mean - 45.0 Variance 15.6 Autocorrelation function (up to lag 5) r1 = 0.80, r2 = .50, rз = .26, r4 = -.10, rs = 0.08: == Estimate the parameters of the model using the method of moments.

Please solve with the step max in 60-90 minutes the chapter actuarial Investigate whether the risk measure Variance principle premium p(L) = μL + ασL2  α = loading factor meets the axioms (T, S, PH and M)

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.

Let X₁, X2, X3, Xn be a random sample with unknown mean EX; = µ, and unknown variance Var(X₂) = o². Suppose that we would like to estimate 0 = μ². We define the estimator as 2 • - (™)² - [ 2x]* Xk to estimate 0. Is an unbiased estimator of ? Why?

An electrochemical engineer has manufactured a new type of fuel cell (a type of battery)which has to undergo testing to prove its duration: the time it takes to go from fullycharged to completely uncharged, under a fixed nominal load. From the computational simulation models she has, the variance of the duration is σ2 = 4 h2 (hours squared)but she wants to estimate the mean duration time μ. To achieve this she is determinedto do the tests multiple times in independent but identical conditions. Can you findwhat is the smallest number of these tests that she has to do in order for her estimatedmean duration to be within ±0.2 h tolerance of the true mean with 95% certainty

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Question 1.2 Consider the function  f (x) = (1/24(x^2 +1)      1 < or =  x < or = 4)          = (0      otherwise)   Calculate P (x = 3) Calculate P (2 < or =  x < or = 3)         Question 1.3 Consider the function  f (x) = (k - x/4      1 < or =  x < or = 3)          = (0      otherwise)   which is being used as a probability density function for a continuous random variable x?   a. Find the value of K b. Find P (x < or = 2.5)

Solve with R, please

Let Y be a random variable with the mgfM(t) = .35e^(−4t) + .1 + .25e^(2t) + .3e^(4t).(a) What is the pmf of Y?(b) Calculate the variance of Y directly from the pmf you determined in part (a).(c) Calculate E[Y^4] directly from the mgf.