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- 1. Suppose the manufacturer's specițications por the length of a certain type of computer cable are likely to be defective as large cable . That is, the probability of randomly producing a cable with length exceeding 1010 millimeters is equal to the probability of producing a cable with length smaler than 990 millimeters · The lo00 ± l0 millimeters. It is known that small cable is just as pro loability that the production procedure meets specițications is known to be 0.97 a What is the proloaoility that a cable selected rand only is too large? b. What is the po bability that a randomy selected calole is larger than 990 millimeters ?If X is a continuous random variable and y = aX + b Prove that E ( Y ) = a E ( X ) + b and V ( Y ) = a² V ( X ) PROBLEM 1. %3DProblem Suppose that X;, i= 1, ., n, are iid Bernoulli(p). (a) Show that the variance of the MLE of p attains the Cramér-Rao Lower Bound. (b) For nz 4, show that the product X,X2X3X4 is an unbiased estimator of p, and use this fact to find the best unbiased estimator of p4.
- 6. Two independent random variables X1=(-1,0,1) and X2=(-1.0) can take the following probability values: P(X1-1)-p, P(X1=0)=2p, P(X1=1)=1-3p, P(X2=-1)=p, P(X2=0)=1-p. Si X=X1+X2 y Y=X1*X2, determine E(XY).A particle is confined in a straight tunnel aligned in east-west direction and it can move randomly by the step size of one or two units. For each movement, the probability is p that the particle will move one unit to the west, the probability is q that the particle will move two units to the east, and the probability is 1 – p – q that the particle will remain at the same place. (0 < p < 1,0 < q < 1,0 < 1- p – q < 1). A movement is independent to another. Calculate the expectation of the position of the particle after n movements, assuming that the position of starting point is 0.B) Let the random variable X have the moment generating function e3t M(t) for -1A) Caused by a third variable B) uncorrelated ) negative effect 0.fb/ orrrte D) direct cause-and-effect 7) The formula for the correlation coefficient r= n (Σxy)-(Σx) (Σν) means 7) n(Ex2)- (Ex)2n(2y2)- (2y) D A) r is the ratio of the covariance to the product of the standard deviation of XY B) r is the ratio of the covariation to the square root of the product of the variation in X and the variation in Y. C)ris the ratio of the covariance to the square root of the product of the variation in X and the variation in Y. D) r is the ratio of the covariation to the product of the standard deviations of X and Y.5. Let Y,, Y2, ., Yn be independent, exponentially distributed random variables with mean 0/2. Show that the variance of the minimum, Y1) = min(Y,, Y2, ...,n), are E(Y1)) Var(Ya)) = and 2n 02 4n²°2) Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function f(x;0)=B (x-4) for x > 4 otherwise Where, ß > 0. a) If f(x; 0) a member of one-parameter regular exponential family. What are the functions h(x), c(0), w (0) and t(x). b) Find the maximum likelihood estimator of the parameter B. c) If the data from this random sample are 8.2, 9.1, 10.6 and 4.9, respectively, what is the maximum likelihood estimate of B?An ecologist is considering two different positions in which to setup his solar energy collector. He takes two identical collector places one in each of the positions and measure the energy in at collected as 10 consecutive days. Days 1 2 3 4 5 6 7 8 9 10 Position X 2.6 5.4 3.2 5.0 4.3 2.7 2.6 3.9 6.0 1.6 POSITION y 2.8 5.1 3.1 6.0 4.6 3.2 2.9 4.5 6.2 2.2 Test whether the difference of mean is significance ? what will be the answer1) Suppose that Y₁, Y2,...,Y₁ is a random sample from a population with probability density function 1,7, f(x)=B³ Find the Maximum Likelihood Estimator of B. 00 elsewhereQuestion 24. (a) Prove that if two continuous random variables X and Y are independent, then the expected value of the product, E(XY) is the product of the marginal expectations. Start the proof with the definition of E(XY). (b) Prove that if two random variables X andY are independent, so are the following functions of the two random variables W = a + bX and T = c + dY. You may use just the expectation operator and the result obtained in part (a).SEE MORE QUESTIONS