4. Show all your work in this exercise - copying the results from the list with...
Please answer A.6.6.: The previous two questions mentioned above are included below: A.6.6. We mentioned in class that the Gamma(, 2) distribution when k is a positive integer is called the Chi-square distribution with k degrees of freedom. From the previous two problems, find the mean, variance, and MGF of the Chi-square distribution with k degrees of freedom. A.6.5. In class we showed that if X ~ Gamma(α, β) then E (X) = aß and uar(X) = αβ2 by using...
00900 Gamma Distribution Exercise. To determine the variance of these estimators, compute the appropriate second derivatives. o? a2 θα β This give a Fisher information matrix /(α, β)::n( ) Inf(a) In r(a) - -a 1(0.19, 5.18) (0.19,5.18) 500 28.983 -0.193 500 NB. ψι(a) d2mT(a)/da2 is known as the trigamma function and is called in R with trigamma. 8/10 00090 Gamma Distribution The inverse matrix 1 0.0422 1.1494 /(α, β)--500 (1.1494 172.5587 Var(a) ~ 8.432 x 10-5 σ& 0.00918 Var(8) 0.3451...
00900 Gamma Distribution Exercise. To determine the variance of these estimators, compute the appropriate second derivatives. o? a2 θα β This give a Fisher information matrix /(α, β)::n( ) Inf(a) In r(a) - -a 1(0.19, 5.18) (0.19,5.18) 500 28.983 -0.193 500 NB. ψι(a) d2mT(a)/da2 is known as the trigamma function and is called in R with trigamma. 8/10 00090 Gamma Distribution The inverse matrix 1 0.0422 1.1494 /(α, β)--500 (1.1494 172.5587 Var(a) ~ 8.432 x 10-5 σ& 0.00918 Var(8) 0.3451...
Show all work! Thank you! kxk-1 4.34 Given the pdf for X is f(x)= 10 0<x<1 otherwise determine E[X] and Var[X]. 1 0<x<1 4.35 Given the pdf for X is f(x)=x. determine E[X] and Var[X]. 10 otherwise' Sections 4.5-4.8 A<x<B 4.36 Given a random variable with pdf f(x)= B-A , determine the MGF for this random variable. 10 otherwise so x50 4.37 Given a random variable with pdf f(x)= betx 0<x , determine the MGF for this random variable. '...
Only 1-6) N(4,) "x.xx be a random sample from variance, respectively. In order to show that and let X and S be sample mean and sample 1. Let and 5 are independent, tollow the steps below. 1-1) Use the change of variable technique =nx-x,- x and show the joint pdf of ,X,,X is (n-1) n- exp f(,x) 20 2a av2 Use Jacobian for n x n variable transformation 1-2) Use the fact that N(u,a n), and show that the conditional...
1] Assume that the helium porosity (in percentage) of coal sample taken from any particular seam is normally distributed with true standard deviation 0.75. (a) Compute a 90% confidence interval for the true average porosity of a certain seam if the average porosity for 20 specimens from the seam was 4.85. (b) How large a sample size is necessary if the width of the 95% interval for the true average is to be 0.30? 2] A commonly used practice of airline...
2. Consider tossing a coin twice. Denote H ="head" and T ="tail" (a) List all outcomes in the sample space S (b) Let X count the number of heads. List all outcomes in the events Ao = {X = 0}, Ai = {X=1 and A2 {X = 2}. Are all the events Ao,A1,A2 mutually exclusive? Explain. (c) Suppose P(H) = 0.6. Find the probability mass function of X: f(x) = P{X =x} (d) Find the cumulative distribution function of X:...
Please show full work and explain your work. Thank you. 4·USEFUL THEORETICAL PROBLEMS Tn (26) Assume the Binomial distribution, with pmf given by: pn(k) n- ptq 12 where p +q (a) Prove that >Pn(k)1 (b) Find kp(k) (c) Find > kp(k) (d) Find kp(R)kp(k) (27) Assume the Poisson distribution, with pmf given by: p(k)exp (-X).1 (c) Find Σk2p(k) a) Prove that > p(k)- 1 (b) Find Σ kp(k) (d) Find | Σ k-p(k) )-1 Σ kp(k) Use Poisson 10 Use...
Please show all work X, be a random sample from the distribution with the probability density function Let A0 and let X, X2, f(x; A) 24xe, x>0. a. Find E(X), where k> -8. Enter a formula below. Use* for multiplication, for divison, ^ for power, lam for A, Gamma for the r function, and pi for the mathematical constant . For example, lam k*Gamma(k/2)/pi means Akr(k/2)/T Ax2 or u =x2. Hint 1: Consider u -e"du Hint 2: I'(a) a 0...
1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X...