a.
Prove that joint density
a.
Explanation of Solution
Calculation:
In Exercise 5.65 the joint density is given as follows:
Where marginal densities of
It is known that the cumulative density function of
Now, substitute
That is,
Hence, the joint probability density function is obtained as follows:
Hence, it is proved that the joint density function of
b.
Evaluate
b.
Answer to Problem 163SE
The joint cumulative density function of
Explanation of Solution
Calculation:
Consider that
Hence, the
Now, substitute
That is,
Hence, the joint cumulative density function of
c.
Obtain the joint density function associated with the distribution function that is obtained in Part (b).
c.
Answer to Problem 163SE
The joint density function is,
Explanation of Solution
From Part (b), the joint cumulative density function of
Hence, the joint probability density function is obtained as follows:
Thus, the joint density function is,
d.
Provide two specific and different joint densities that yield marginal densities for
d.
Explanation of Solution
Calculation:
From Part (c), the density function is obtained as follows:
Where marginal densities of
The marginal density function does not depend on the values of
Thus, as
Consider
Consider
Hence, these two specific and different joint densities yield marginal densities for
Want to see more full solutions like this?
Chapter 5 Solutions
Mathematical Statistics with Applications
- Find two nonnegative number x and y for which x+3y=30, such that x2y is maximized.arrow_forwardPlease help asap with this math theory question (: Consider the delta property and the Von Neumann Morgenstern axioms for the following: If you have a risk premium for Gamble A with a probability 1-p of winning r1 and probability p of winning r2. It's y' for Gamble A' with probability p of winning r2 + c, probability 1-p of winning r1+ c. What is y' as a function of y?arrow_forwardSamples of 2 units will be selected from the population whose distribution is given below, with a return and order of importance (W / R, O / I). a)Show that x̄ is a neutral estimator. b)Find V(x̄). Show that V(x̄) is equals to ϭ²/n. c)Find E(s²). Show that E(s²) is equal to ϭ². (Hint: W/R=With Replacement, O/I=Order Important)arrow_forward
- Q1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 0arrow_forward1. Find the uncertainty equation for the following function f, assuming x, y, and z are variables. |2x* – 3xy – 5y* f(x,y,z) =, %3D 3zarrow_forwardLet X1, . . . , Xn, . . . ∼ iid Bern(θ). Consider the Bayes estimator under squared error loss with the Unif(0,1) prior. Show that this estimator is consistent.arrow_forwardWhich of the following sets are linearly independent in p3? %3D Sz = {(1.21). (1.-1,0). (1., 1,1) S3 = ((0,0.–1)- (0,1.0). (-20.0) OA. S.S3 O B. S3 %3D OC.S.S2 O D. S,,S3 OES,arrow_forward4. Let S = {-1,0,2,4,7}. Find f(S) if а. f(x) — 1 %3D b. f(x) 3D 2х+ 1 c. f(x) = [*/5] d. f(x) = [x* + 1/3]arrow_forward2. Consider the function x < 0, 0 x/2 0 < x < 1, k+ (1 k) (x − 1)² 1 ≤ x < 2, 1 x ≥ 2. (a) State the 4 properties of a distribution function. Fx (x) =arrow_forwardarrow_back_iosarrow_forward_ios
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,