Exercise 6 W, X, Y and Z are independent standard normal random variables. b) P(4X +3Y < Z+W) = }; c) E(4X +3Y - 2Z² - W² + 8) = 8; g) P(|min(X,Y)| < 1) = 0.6826;
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- Let X(n,p) denote binomial random variable with parameter n and p. so that P(X(n,p) > ( greater equal) x) = summation P(X=i) ( summation limit i=x to n ) if W=X(n+1,p) and Y=X(n,p) explay why W>(greater equal "st" Y) See attached for the correct format.Let � be a continuous random variable distributed by �(11,1.2), find:A) �(�≥8.89)= Keep your answer in 4 decimal places.B) �(�>11.97)= Keep your answer in 4 decimal places.**Notes: DO NOT round any z-score.a) Let X₁, X₂, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) you and Y = -√n. VR i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal rando... variable. iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X₁, X₂, X, and Y₁, Y₂.... Y be random samples from populations with moment generating functions Mx, (t) = eat+t² and My (t) = (-), respectively. 25 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ X3 + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)2> 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-Hy ≥10) <0.04? QUESTION A
- Suppose the distribution of some random variable X is modeled as Negative binomial with parameters, p and r, where p is the probability of success and r is the total number of success over the trials. Find the method of moment (MM) estimators for (p,r), in closed algebraic form expressions based on a random sample of size n, X1, X2,...,Xn.Show that two random variables X, and X, with joint pdf. Sx, x, (*1 »*2) = 16 Xl< 4, 2< X2<4 are independent and orthogonal.Let Z be a random variable with E(Z) = 12 and Var(Z) = 5. Based on the statement above, determine which of the following statements are true and false. Show complete solution. a) E(3Z + 10) = 46 b) E(Z^2) = 160 c) Var(10)=10
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