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**Q1:**Let X denote the amount of time for which a book on 2-hour reserve at a college library ischecked out by a randomly selected student and suppose that X has density function f(x)=\left\{\begin{array}{ll} 0.5 x & 0 \leq x \leq 2 \\ 0 & \text { otherwise } \end{array}\right. calculate the following probabilities: P(0. 5 < X< 1.5)See Answer**Q2:**Assume that corporate bond ratings follow a time invariant Markov process with the abovetransition matrix, , and states {AAA, AA, A, BBB, BB, B,CCC, Default}, represented bythe state s E {1, ..., 8}, where s = 4, e.g., represents a BBB rating. ) What is the probability that a bond over two years moves from an AAA rating to Default,with the above transition matrix? b) What is the long-term (stationary) distribution, p of ratings for a corporate bond withthe above transition matrix (assuming that it has a very long maturity sate)?See Answer**Q3:**7. Explain what the phrase " 99% confident" means. A) 99% of the sample means from similar samples fall within the interval. B) 99% of the similarly constructed intervals would contain the value of the sample mean. C) In repeated sampling, 99% of the intervals constructed would contain u. D) 99% of the population values will fall within the interval.See Answer**Q4:**Use the code below to generate 1000 samples of the random variables X, Y, and Z. x-randn(1,1000); y=randn(1,1000); Z-x+y; Applying equation 3-25 on page 133 of the text, compute the correlation coefficient for the following cases;а. p(x,y), p(x,x), and p(x,z). Do not use the MATLAB function coefficient Provide your code and the results out out of MATLAB. Explain the results for each output in MATLAB. b. Using a criteria of p>.75 for significant correlation, determine if any of the cases above in part a. result in significant correlation. If we assume that the case where p<.75 is uncorrelated, assess the three cases as statistically independent or statistically dependent?See Answer**Q5:**Problem 6-3.2. But change problem to say sine instead of cosine. In Part (a) Change problem to"is this process wide sense stationary?" You may ignore the question about whether it is say"ergodic "See Answer**Q6:**. Let X(t) and Y(t) be zero-mean independent wide sense stationary random processes with autocorrelation functions R_{x}(\tau)=100 e^{-|\tau|} \cos (4 \pi) R_{x}(\tau)=50 e^{-\tau^{2}} Define a new random process Z(t)=X(t-1)+Y(t-2). Find the auto correlation function of Z(t). Is Z(t) a wide sense stationary random process? Find the correlation function between Z(t) and Y(t).See Answer**Q7:**4. Problem 6-4.1. But use the sample values for x_k below instead of the one in the book. See Answer**Q8:**The continuous random variable X has the following cumulative distribution function: Estimate the following probabilities: a) P(X < 50) P(X40)See Answer**Q9:**For the tabulated values of the discrete random variable X and the correspondingprobabilities, estimate the following: ) ΡX< 2) P(X > -2) O P(-1 <X< 1) • P(X <-1 or X = 2)See Answer**Q10:**There is a chance that a bit transmitted through a digital transmission channel is received in error. Let X equal the number of bits in error in the next four bits transmitted. The possible values for X are {0,1,2,3,4}. Based on a model for the errors that is presented in the following section, probabilities for these values will be determined. Suppose that the probabilities are P(X = 0) = 0.6561 P(X = 2) = 0.0486 P(X = 4) = 0.0001 P(X = 1) = 0.2916%3D Р(X%—D 3) 3 0.0036%3D Estimate the mean, variance and standard deviation for the random variable X.See Answer**Q11:**Suppose the reaction temperature X (in degrees centigrade) in a certain chemical process hasa uniform distribution with A = -5 and B = 5. Compute the following probabilities. P(X<0) Р(-2 <X <2)See Answer**Q12:**Problem 6-2.1. Problem 6-3.2. But change problem to say sine instead of cosine. In Part (a) Change problem to say “is this process wide sense stationary?" You may ignore the question about whether it isSee Answer**Q13:**Let X and Y be random variables with positive variance. The correlation of X and Y is defined as \rho(X, Y)=\frac{\operatorname{cov}(X, Y)}{\sqrt{V(X) V(Y)}} =) Using Exercise 17(c), show that 0 \leq V\left(\frac{X}{\sigma(X)}+\frac{Y}{\sigma(Y)}\right)=2(1+\rho(X, Y)) (b) Now show that 0 \leq V\left(\frac{X}{\sigma(X)}-\frac{Y}{\sigma(Y)}\right)=2(1-\rho(X, Y)) Using (a) and (b), show that -1 < p(X,Y) <1.See Answer**Q14:**Let X and Y be random variables. The covariance Cov(X, Y) is defined by(see Exercise 6.2.23) cov(X, Y) = E((X – 4(X))(Y – µ(Y))) .- (a) Show that cov(X, Y) = E(XY) – E(X)E(Y).%3D (b) Using (a), show that cov(X,Y) = 0, if X and Y are independent. (Caution: the converse is not always true.) (c) Show that V(X +Y) V(X)+V(Y)+2cov(X,Y).See Answer**Q15:**Let X and Y be independent random variables with uniform densities in [0, 1].Let Z = X +Y and W = X - Y. Find p(X,Y) (see Exercise 18). O P(X, Z). p(Y, W). p(Z, W).See Answer**Q16:**24 Let X be a random variable with density function fx. The mode of X is the value M for which f(M) is maximum. Then values of X near M are most likely to occur. Find M if X is distributed normally or exponentially, as in Exercise 22. What happens if X is distributed uniformly?See Answer**Q17:**26 Bridies' Bearing Works manufactures bearing shafts whose diameters are nor-mally distributed with parameters µ = 1, o = .002. The buyer's specificationsrequire these diameters to be 1.000 +.003 cm. What fraction of the manu-facturer's shafts are likely to be rejected? If the manufacturer improves herquality control, she can reduce the value of o. What value of o will ensurethat no more than 1 percent of her shafts are likely to be rejected?See Answer**Q18:**Consider the circuit shown in the figure above where the probability that each device functions correctly is provided. Assume that all devices fail independently. What is the probability of two or fewer failed devices? See Answer

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