(c) In 1955, C.W. Topp and F.C. Leone introduced a number of distributions in the context of the statistical modelling of the reliability of electronic components in engineering. One of these distributions has probability density function (p.d.f.) given by f(x) = 4x(1x) (2 - x) on the range 0 < x < 1. (i) Verify, by integration, that [ 4x(1 − x)(2 − x) dx : 4x(1-x)(2-x) dx = x²(2 − x)² + c, where c is an arbitrary constant. (ii) Explain why the p.d.f. suggested by Topp and Leone is a valid p.d.f. (iii) What is the c.d.f. associated with this p.d.f.?
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- If X has probability density function f(x) = X/32 on [0, 8], find the expected value and the standard deviation of X. Give your answer either as a irreducible fraction or a decimal number accurate to 2 decimal places. E(X) = std(X) =For what value of c is the function f(x) = {c/x³ if x ≥ 1 {gx3 otherwise a probability density function?Suppose that we have an independent and identically distributed sample from some distribution family G defined by an unknown parameter value 1. We are testing the hypotheses H,:A = 1 versus Ha:A = 2. Our test statistic is T, and let T have an Expo(1) distribution. The Expo(1) probability density function (pdf) and cumulative distribution function (cdf), respectively, are fr(t) = le-a for t >0 Fr(t) = 1- e-dt for t > 0 The rejection region for this hypothesis test is {T < 0.5}.
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