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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: The continuous random variable X has the following cumulative distribution function: Estimate the following probabilities: a) P(X < 50) P(X40)See Answer
Q5: 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
Q6: 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
Q7: 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
Q8: 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
Q9: 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
Q10: 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
Q11: 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
Q12: 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
Q13: 8. What assumption is necessary for the confidence interval analysis to work properly? A) The population that we are sampling from needs to be approximately normally distributed. B) The Central Limit theorem guarantees that no assumptions about the population are necessary. C) The sampling distribution of the sample mean needs to be approximately normally distributed. D) The population that we are sampling from needs to be at -distribution with n-1 degrees of freedom.See Answer
Q14: The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows: Calculate the mean and variance for X. Show all steps. Do it without using special functions on calculator.See Answer
Q15: The random variable Y is related to the random variable X by the relationship; Y=2•X + 6. Given the PDF for the random variable X is f_{x}(x)=1 / 2 ; \quad 0<X \leq 2 find the probability density function f,(y) for therandom variable Y.Remember that the range for Y willbe [6,10] given the relationship above and the range[0,2] of the random variable X. Now compute the mean of the random variable Y. for theAfter you have computed the theoretical meanrandom variable Y, use the matlab code below to verifyDo they agree?your solution. X=2*rand (1,100000); %X is a uniform RV between 0 and 2Y=2*X+6; % Y is a RV formed from X mean (Y) % This is the mean of the RV YSee Answer
Q16: In a conference, we have 15 people with college degree 20 people who have job 40 people with no college degree 15 people with no college degree having job Use Venn Diagram to find, how many people are in this conference? O 75 O 30 D 55See Answer
Q17: The table below shows the observed values of X and Y. To determine the regression line, you haveto: ] Complete the Calibration table (Model Calibration). 》 Establish a mathematical relationship between X and Y. See Answer
Q18:Five hundred ball bearings have a mean weight of 5.02 oz and a standard deviation of 0.30 oz. Find the probability that a random sample of 100 ball bearings chosen from this group will have a combined weight of more than 510 oz.See Answer

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