9)- Suppose the random variable y has a Weibull density function with a = 4 and ß = 100. %3D 1. Find F(5). 2. Find P(y 2 3) 3. Find u and o

2. Let X be a Chi-squared random variable with density function fx(x) = exp x > 0. 2 Suppose Y is independent of, and has the same distribution as, X.

21. An instrument is used to measure very small concentrations, X, of a certain chemical in soil samples. Suppose that the values of X in those soils in which the chemical 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 determined to be present. At very low concentrations, however, the chemical may fail to be detected even if it is present. This phenomenon is modeled by assuming that if the concentration is x, the chemical is detected with probability R(x). Let Y denote the concentration of a chemical in a soil in which it has been determined to be present. Show that the density function of Y is g(y): = R(y) f (y) R(x) f(x) dx

2. Let X be a Chi-squared random variable with density function a fx (x) = exp 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 %3D fz(z) = 2a exp z > 0. Hence state the value of a. (b) Let V = X +Y and W = X/Y. Show that the joint density of V and W is given by 1 fv,w(v, w) = v, w > 0. ev/2 2T /w(w +1)

8. Suppose that Y1, Y2, and Y3 denote a random sample from an population with an exponential distribution with a density function of f (æ) = !eō for y > 0. Determine if 0 = 1+212 is a biased or unbiased estimator of 0. Yı+2Y2 %3D Justify your answer on your work paper. O Unbiased O Biased

2. Let X be a random variable with the following density. c(1 – 2)3 if r € (0, 1), otherwise. f(x) = (i) Find the constant c and then E(X) when X has the above density. (ii) Find E(X²) when X has the above density. • (iii) Find Std(X) when X has the above density.

7. The joint probability density function of two dimensional random variable(X,Y) is given 8. by f(x, y) = 7 xy, 1sxsys2 oitoupslo %3D = 0, elsewhere Find the marginal density functions of X and Y.

a) What is the value of C. b) What is the variance of X.

36. QUESTION: Let X have the density f(x) = 2x if 0 < r <1 and f(x) = 0 otherwise. Show that X has the mean 2/3 and the variance 1/18. Find the mean and the variance of the random variable Y = -2X + 3.