1. Given a random sample X, from a population with density f(x; 8), show that maximizing the likelihood function L(0x) is equivalent to maximizing In L(0|x).
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- The density function of a random variable is given as fx(x) = ae bx x20. Find the characteristic function and the first two moments.Let X1, X2, ..., X50 be a random sample of size 50 from a distribution with density for 0inclue the varianceThe random variable X models the loss in thousands of dollars due to a fire in a commercial building. Its density function is f(x) = 0.005*(20-x) for 0 being less than or equal to x and x being less than or equal to 20 and f(x)= 0 otherwise. Given that a loss due to the fire exceeds $8000, find the probability the loss exceeds $16000.The life length, T, of an electric component is a random variable with hazard rate function h(t) = (60+t) -;7>0. Obtain (i) Cumulative Hazard Rate Function, (ii) Reliability Function, (iii) Density Function and (iv) Conditional Reliability Function for T.An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the chemical is first determinedto be present. At very low concentrations, however, the chemical may fail tobe detected even if it is present. This phenomenon is modeled by assuming thatif the concentration is x, the chemical is detected with probability R(x). Let Ydenote the concentration of a chemical in a soil in which it has been determinedto be present. Show that the density function of Y isg(y) = R(y) f (y)/An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the…Find the maximum likelihood estimator of the unknown parameter where X₁, X2,..., X₂ is a sample 0 from the distribution whose density function is fx(x) = { e-(1-0) if x > 0 otherwise.3. The storm runoff X (in cubie meters per second, ems) can be modeled by a random variable with the following probability density function (PDF): fa(x) = c (x -) for 0The probability density function of a continuous random variable is given by 1 1 4 f (x) = - 4 for x > 0. a) Find P(X> 5). b) Find the mean of X. c) Find the CDF (cumulative distribution function). d) Use the CDF to find the median of X.The time it takes to repair a personal computer is a random variable whose density, in hours, is given by E 0Let f, g be probability densities such supp(f) c supp(g). show yes X1, ... , Xn X1,..., X, iid. L(g) then f(X;) W; = g(X;) is hopeful 1. Argue why the weights need to be re-normalized even when the normalization constants of the densities f and g are known.Let X denote the temperature at which a certain chemical reaction takes place. Suppose that X has pdf f(x)={1/9(4-x^2) -1<=x<=2 0 otherwise a) Compute P(0≤ X ≤1)b) Obtain E(x) and Variance of XSEE MORE QUESTIONSRecommended textbooks for youLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
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