水文统计习题1英文版

Test 1
1、If A, B and C are events, which of the following relationships are true? (a) ( A B) ( A C ) A ( B C ) (c) A B A B (e) ( A B) ( B C ) 2、Suppose that A, B, and C are events such that P ( A) P( B) P(C ) 1/ 4 , (b) ( A B) ( A B ) B (d) ( A B) C A B C
versus
H1 : 30.0 . The variance is assumed to
ቤተ መጻሕፍቲ ባይዱ
be unknown since the suggested climatological changes may also affect the variability of the rainfall, and hence past records on the variance are not meaningful.
affecting, among other things, the annual precipitation. Let us assume that the past 8 years have yielded the following annual precipitation (inches): 34.1, 33.7, 27.4, 31.1, 30.9, 35.2, 28.4, 32.1 It is hypothesized that in fact the annual rainfall has increased. Specifically we want to test H 0 : 30.0
f ( x) 8 / x 3 , x 2;
g ( y ) 2 y, 0 y 1.
(a) Find the pdf of Z XY . (b) Obtain E ( Z ) in two ways: (i) using the pdf of Z as obtained in (a). (ii) directly, without using the pdf of Z. 10 、 If X, Y, and Z are uncorrelated random variables with standard deviations 5, 12, and 9, respectively and if U X Y and V Y Z , evaluate the correlation
8、Suppose that the continuous two-dimensional random variable(X,Y) is uniformly distributed over the square whose vertices are (1,0), (0,1) (-1,0), and (0,-1). Find the marginal pdf’s of X and of Y. 9、Suppose that X and Y are independent random variables with the following pdf’s:
coefficient between U and V. 11、 Particles are emitted from a radioactive source. Suppose that the number of such particles emitted during a one-hour period has a Poisson distribution with parameter . A counting device is used to record the number of such particles emitted. If more than 30 particles arrive during any one-hour period, the recording device is incapable of keeping track of the excess and simply records 30. If Y is the random variable defined as the number of particles recorded by the counting device, obtain the probability distribution of Y. 12 、 A distribution closely related to the normal distribution is the lognormal distribution. Suppose that X is normal distributed with mean and variance 2 . Let Y e X . Then Y has the lognormal distribution. (That is, Y is lognormal if and only if ln Y is normal.) Find the pdf of Y. Note: The following random variables may be represented by the above distribution: the diameter of small particles after a crushing process, the size of an organism subject to a number of small impulses, and the life length of certain items. 13、Assume that the number of accidents in a factory may be represented by a Poisson process averaging 2 accidents per week. What is the probability that (a) the time from one accident to the next will be more than 3 days; (b) the time from one accident to the third accident will be more than a week? [Hint: in (a), let T=time (in days) and compute P (T>3).] 14 、 Independent samples of size 10 and 15 are taken from a normally distributed random variable with expectation 20 and variance 3. What is the probability that the means of the two samples differ(in absolute value) by more than 0.3? 15、Suppose that X has a Weibull distribution with pdf
f ( x) ( ) x 1e x ,

x0.
Suppose that is known. Find the ML estimate of based on a sample of size n. 16 、 Suppose that X is normally distributed. A random sample of size 4 is obtained and X the sample mean is computed. If the sum of squares of the deviations of these 4 measurements from X equals 48, obtain a 95 percent (two-sided) confidence interval for E(X) in terms of X . 17、Suppose that X, the annual rainfall at a certain locality, is normally distributed with E(X) = 30.0 inches. (This value has been established from a long history of weather data.) In recent years, certain climatological changes seem to be evident
P ( A B) P(C B) 0 , and P ( A C ) 1/ 8 .Evaluate the probability that at
least one of the events A, B, or C occurs. 3、In a bolt factory, machines A, B, and C manufacture 25, 35, and 40 percent of the total output, respectively. Of their outputs, 5, 4, and 2 percent, respectively, are defective bolts. A bolt is chosen at random and found to be defective. What is the probability that the bolt came from machine A? B? C? 4、A die is thrown n times. What is the probability that “6” comes up at least once in the n throws? 5、Four radio signals are emitted successively. If the reception of any one signal is independent of the reception of another and if these probabilities are 0.1, 0.2, 0.3, and 0.4, respectively, compute the probability that k signals will be received for k = 0, 1, 2, 3, 4. 6、The diameter on an electric cable, say X, is assumed to be a continuous random variable with pdf f ( x) 6 x(1 x), 0 x 1 . 7、Suppose that the continuous random variable X has pdf f ( x) e x , x 0 . Find the pdf of the following random variables: (a) Y X 3 ; (b) Z 3 /( X 1) 2 .