a.
Prove that the marginal distribution of
a.
Explanation of Solution
Calculation:
Consider that
Then, the marginal probability
Exponential distribution:
A random variable Y is said to follow the exponential distribution with mean 1, if and only if the probability density function of Y is
Consider that
Thus, using the joint probability density functions of
Thus, it is proved that the marginal distribution of
b.
Find the marginal distribution of
b.
Answer to Problem 65E
The marginal distribution of
Explanation of Solution
Calculation:
Consider that
Thus, using the joint probability density functions of
Thus, the marginal distribution of
c.
Prove that
c.
Explanation of Solution
The two continuous random variables
Substitute
That is,
From Part (a) and Part (b), the marginal distribution of
Here, the ranges of
That is,
Thus, it is proved that
Want to see more full solutions like this?
Chapter 5 Solutions
MATH.STATISTICS W/APPL.-ACCESS
- Suppose the joint probability density of X and Y is fX,Y (x, y) = 3y 2 with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and zero everywhere else. 1. Compute E[X|Y = y]. 2. Compute E[X3 + X|X < .5]arrow_forwardFind a value of k that will make f a probability density functionon the indicated interval.ƒ(x) = kx; [0, 3]arrow_forwardFind a value of k that will make f a probability density function on the indicated interval. ƒ(x) = kx; [2, 4]arrow_forward
- Suppose that the random variables X,Y, and Z have the joint probability density function f(x,y,z) = 8xyz for 0<x<1, 0<y<1, and 0<z<1. Determine P(X<0.7).arrow_forwardSuppose 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_forwardThe PDF of a continuous random variable X is as follows: f(X)= c(4x2 - 2x2) 0<* x <* 2 (*less or equal to) a. For this to be a proper density function, what must be the value of c ?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