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- The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.Let X₁, X2, X3, X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x) = (0, -, if 0An 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…If a dealer's profit, in units of $3000, on a new automobile can be looked upon as a random variable X having the density function below, find the average profit per automobile. f(x) = 1 (7-x), 0Q4- If the density function of X is 1 fx(x): (x + 1)2* > Find the probability density function of Y = X+1 using both Transformation Method and Distribution Function Method.A system consisting of one original unit plus a spare can function for a random amount of time X. If the density of X is given (in units of months) by f(x) = Cxe-x/2 x > 0 x ≤0 least 5 months? Yanıt: what is the probability that the system functions for atSEE MORE QUESTIONSRecommended textbooks for youAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage