1.Suppose that X and Yare two continuous random variables with the joint probability density function /(x.y) - x + 2y for 0 ＜x＜2 and 0 ＜y ＜ 2. The covariance of Xand Y is________.

7.Find the det(A) = ?, det(A x B)=?, and =?.

8.f = , plot the curve. Hin. Through the quadratic form, transform it to principal axes, and express [x1  x2] in terms of the new coordinate vector [y1 y2]. Please try specifying each dimension of your plotted conic section. (10 分)

9. Find the complex Fourier Series of f(x)=ex if-p＜x ＜pandf(x+2p)=T(x).(10分)

10. Find the trigonometric polynomial F(x) of the form:for which the square error with respect to the given f(x) = x2 on the interval -π ＜x ＜ π is minimum. Compute the minimum value for N = 1, 2, ..., 5. (10 分)

2.A random variable X has a mean p = 5 and variance σ2 = 36. Based on Chebyshev's inequality, a lower bound on Pr(-7＜X＜ 17)is____________.

3.Suppose that X is a father's height, Y is a mother's height, Z is a child's ultimate height (as an adult), and G is delined as: G = I if the child is a boy and G = -I ifthe child is a girl. Let M = (X+ Y)/2. Suppose that, conditional on known values of G, X, and Y, Z has a continuous uniform distribution with range [M -5 +3G, M + 5 + 3G]. If a girl's father is 4 inches taller than her mother, the probability that she will grow to be taller than her mother is__________.

4.A manufacturer of car tires claims that his tires will last, on the average, 4 years with a standard deviation of 1 year. Given that 5 of these tires have lifetimes of 2.1, 2.2, 3.0, 3.6, 4.1 years, if someone wants to test whether his tires have a standard deviation of1 year, the figure of the appropriate test statistic is equal to__________.

5.You are going to carry out an experiment to determine whether surface finish has an effect on the endurance limit of steel. It is expected that polishing increases the average endurance limit. In such an experiment, you want to detect that polishing fails to have an effect with a probability of 0.99 (i.(E), a = 0.01) and you also want to detect a change in the average endurance limit of 700 units by a probability of at least 0.9. If it is known that the standard deviation of the endurance limit of the steel is 420 units, then the sample size that would be needed to perform this experiment is____________ . Assume that for a = 0.01, Za = 2.5 and forβ -0.10, 2β- 1.5.

6.How large a sample must be taken in order that you are 95 percent certain that is within 0.5 of the standard deviation of the mean?____________.

7. Ten trials are conducted of a given system with the following results: S, S, F, S, S, S, F, S, S, S ("S" represents success and "IF" represents failure), assuming that the trials are independent of each other. The maximum likelihood estimate of the probability of successful operation is___________.