Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of X is f(x; 0) = [ (0 + 1)x® where -1 < 8. A random sample of ten students yields data x₁ = 0.86, x₂ = 0.92, x3 = 0.95, x4 = 0.94, X5 = 0.45, x6 = 0.65, x = 0.73, xg = 0.90, x9 = 0.79, X10 = 0.79. (a) Use the method of moments to obtain an estimator of Ⓒ 1 (1 + X) 1 0 (x - 1) - 0 (1 + 7) - 1 1 0 (1-7) - - 2 -n ΣIn(X) Eln(X;) n 2 1 0 (7-1) - Compute the estimate for this data. (Round your answer to two decimal places.) 0≤x≤ 1 otherwise 1 (b) Obtain the maximum likelihood estimator of 0. Eln(X;) n ΣIn(X;) -n - 1 1 n ΣIn(X;) Compute the estimate for the given data. (Round your answer to two decimal places.)

Suppose that the lifetime in years, X,  of a type of electrical switch follows the distribution f(x)=12e−x2f(x)=12e−x2for x>0x>0. What type of continuous distribution is this and what is the average lifetime of one of these switches?

(a) Determine the value of k for a valid pdf.(b) Complete the cdf of X .(c) What is the expected total deduction on a randomly chosen form?(d) Let Y = ln(X /5), where ln(·) is natural logarithm. Determine the pdf as wellas the cdf of Y , and further identify which common distribution that Y follows?

Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 43 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.50 ml/kg for the distribution of blood plasma. (a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.) lower limit     1 upper limit     2 margin of error     3 (b) What conditions are necessary for your calculations? (Select all that apply.) n is large the distribution of weights is normal σ is known the distribution of weights is uniform σ is unknown (c) Interpret your results in the context of this problem. 1% of the intervals created using…