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
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Chapter 5 Solutions
Mathematical Statistics with Applications
- There are only two states of the world, when a person is well with probability (1-p) and ill with probability p, where (1-p)= 1/3 and p = 2/3. Consider Adam who has utility function U = (Y1, Y2,1 – p,p) = Y,"-P)Y?, where Y; is the income and i =1 is well and i = 2 is ill. When Adam is well, he earns $1000, but when he is ill, he losses $300 in health expenditures and earnings such that Y2 = $700. What is the maximum total premium that Adam is still willing-to-pay? (1-р).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_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_forward
- If X1, X2, ... , Xn constitute a random sample of size nfrom a geometric population, show that Y = X1 + X2 +···+ Xn is a sufficient estimator of the parameter θ.arrow_forward7 The function f(t) -7t/3 3 models a continuous probability for 0arrow_forwardSuppose we want to predict a continuous response Y. The true conditional expectation of Y is given by, E(Y | X, Z) = Bo+B1X+B2Z Here Z is a continuous but an unobserved predictor and X is an observed continuous predictor. From the real life data we can only observe y₁, y2, ,..., Yn and x₁, x2,..., n and fit the following model, ŷi = ß0 + Â₁xi Then answer the following problems, (a) The estimator B₁ is a biased estimator. Calculate the bias of the estimator. That is calculate B₁ - E₁). Show the detailed derivations. (b) |Calculate the Var(8₁ | X, Z) using the fitted model under the constant variance assumption. Show the detailed derivations.arrow_forwardQ1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 0arrow_forwardb) c) d)arrow_forwardchoose correct options..hand written plzarrow_forwardchoose correct options..hand written plzarrow_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_forward7. Assume that X is a nonnegative, integer-valued random variable with p.g.f. G(z). Let G(¹) (z) denote the nth derivative of G. Let pk = P{X = k} for k = 0, 1, 2,.... Express each of the following as a function of G. (a) E[X]. (b) E[X(X - 1)]. (c) E[X(X - 1)(x - 2)]. (d) Var(X). (e) po. (f) P1. (g) Pn for n = 0, 1, 2,.... (h) Suppose an archer takes X shots at a target. Assume the shots are inde- pendent, and the archer hits the bullseye on any attempt with probability p. What is the probability that the archer is hits the bullseye on all X attempts? (i) Suppose the effective interest rate per period is i%. What is the present value of one dollar received at time X?arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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