a.
Find the density
a.
Answer to Problem 76SE
The density function of
Explanation of Solution
Calculation:
From Theorem 6.5, the density function for kth order statistic,
It is known that, the density function of Uniform distribution over the interval
The density function for
b.
Find the value of
b.
Answer to Problem 76SE
The value of
Explanation of Solution
Calculation:
Consider,
This function
Therefore,
c.
Find the value of
c.
Answer to Problem 76SE
The value of
Explanation of Solution
Calculation:
Consider,
Therefore,
d.
Find the mean difference between two successive order statistics,
d.
Answer to Problem 76SE
The mean difference between two successive order statistics,
The expected order statistics are equally spaced.
Explanation of Solution
Calculation:
Consider,
This function is a constant for all k. Thus, the expected order statistics are equally spaced.
Want to see more full solutions like this?
Chapter 6 Solutions
EBK MATHEMATICAL STATISTICS WITH APPLIC
- Suppose that the random variables X and Y have a joint density function f(x,y).prove that Cov(X,Y)=0 if E(X|Y=y) does not depend on yarrow_forwardLet X be a continuous random variable with density function f(x) = 2x, 0 ≤ x ≤ 1. Find the moment-generating function of X, M(t), and verify that E(X) = M′(0) and that E(X2) = M′′(0).arrow_forwardFor random variables X and Y with joint density function f(x,y) = 6e-2x-3y. (x,y > 0) and f(x,y) = 0, otherwise, find: a) P(X <= x, Y <= y) b) fx(x) c) fy(y) d) Are X and Y independent? Give a reason for your answer.arrow_forward
- If X and Y are independent exponential random variables, each having parameter λ.(a) Find the joint density function of U = X + Y by using the convolution of fX and fY .(b) Find the joint density function of V = X − Y by using the method of transformation.(c) Are U and V independent?arrow_forwardUse the rejection method to generate a random variable having density function f(x) = kx1⁄2e−x , x > 0 where k =1/Γ (3/2) =2√πarrow_forwardLet f(x, y) = C/8, 0 ≤ y ≤ 4, y ≤ x ≤ y + 2, be the joint pdf of X and Ya) Find the value of C so that f(x,y) is a valid joint p.d.fb) Find the marginal probability density function of X, fX(x)c) Compute μX , μY , Cov(X, Y), and ρ.arrow_forward
- Let X and Y be two random variables with joint density function f(x,y) = (3 − x + 2y) / 60, for 1 < x < 3, 0 < y < 5. Is P(X > 2, Y < 3) equal to P(X > 2) × P(Y < 3)?arrow_forwardThe random vector (X, Y ) has the following joint probability density function:f(X,Y )(x, y) = 4xye^−(x2+y2), x > 0, y > 0,0 , otherwiseLet Z =√(X^2 + Y ^2) . Find the probability density of the random variable Z.arrow_forwardLet X and Y be two independent random variables with densities fX(x) = e^(-x), for x>0 and fY(y) = e^y, for y<0, respectively. Determine the density of X+Y.arrow_forward
- Let X be a random variable with density function f(x) = cx−3, if x ≥ 1, 0 otherwise. a) Find c.b) Find P (3 < X ≤ 6).c) What is P(X = 3)?arrow_forwardIf X is uniformly distributed over (0,1), find the density function of Y = eXarrow_forwardLet (X, Y ) be a random vector with joint density function of the form:f(X,Y )(x, y) = 24xy, 0 < x < 1, 0 < y < 1, x + y < 10 , otherwise(c)Compute the conditional probability P(X < 1/6|Y=1/2) and the conditional expectation E(X | Y = y)arrow_forward
- MATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th...StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C...StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage Learning
- Elementary Statistics: Picturing the World (7th E...StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman