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- 2. Let X be a Chi-squared random variable with density function fx(x) = exp (-;), a x > 0. Suppose Y is independent of, and has the same distribution as, X. (a) Let Z = VX. Show that the density function of Z is given by Sale) = 2a exp (-) z > 0. Hence state the value of a.Suppose that X1, X2, and X3 are i.I.d random variables each with density function f(x)= {((1/4)x^3) if 0 < x < 2 { 0 otherwise Let the random variable M be the max of X1,X2, and X3. What is the E(M)?A 10 A continuous random variable that can assume values between X = 2 and X = 5 has a density function given by fx (x) = K (1 +x). Find p (3 < X < 4)
- Suppose that the continuous random variable X has CDF Fx(X) = {(x-2)/x , x>2 and 0, x=<2} a. Determine, and sketch, the pdf (probability density function) of X. b. Find the mean and variance of X c. Determine the pdf of the random variable Y=X^2Let the density function of a random variable X be given by Sz(x) = 0x*-!, 0sxSI Compute the sufficient estimator of 0 E log (.X,) Ex; a) b) c) Elog (X, ) E-log (X,) d)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…
- Let x be a random variable with a density function I (2) = { "0, 6x (1 – a), 0 < x < 1 elsewhere By finding the fırst and second moments, calculate the variance of x [1] (answer correct to 1 dp)Consider the probability density function P. (x) = a e-b lxl where X is the random variable which assum es all the values from - ∞ to o. Find CDr ) 4.Let Yi,..., Yn denote a random sample from the density function given by fy (ul0) = ry"-le-"10, for 0 > 0, y > 0. where r is a positive constant. Find the MLE of 0. Is the estimator an MVUE for 0?
- Suppose that X is a non-negative random variable with continuous density function fx (z) and a > 0. Show that P + P(X> a) ≤ EX E(X)Suppose that X1, X2, . . . , X7 are i.i.d. random variables, each with density function f(x)= {2(1-x) if 0 < x < 1 { 0 otherwise Let the random variable M be the maximum of X1,X2,...,X7. What is the probability that M< 9/10? Round the answer to nearest hundredth.Suppose that two continuous random variables X and Y have joint probability density function fxy = 1sxs2,0sy<3 elsewhere Find the strength of the relationship and interpret the findings.