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Introduction To Probability And Statistics For Science Engineering And Finance 1st Edition Walter A. Rosenkrantz - Solutions

Problem 8.11 Suppose we use a sample of size n = 25 taken from a normal distribution with σ = 10 to test the hypothesis H0: µ = 50 against H1 : µ= 50. (a) Construct a test of H0 with significance level α = 0.05 by constructing a 95% confidence interval for µ. (b) Describe the rejection region
Problem 8.10 The concentration of an impurity in an ore, measured as a percentage, is assumed to be N(µ, 0.052). The ore cannot be used without additional processing if µ ≥ 0.1%. To determine whether or not the ore is usable, a random sample of size n = 9 is drawn and the ore is declared
Problem 8.9 The breaking strengths of steel wires used in elevator cables are assumed to come from a normal distribution with known σ = 400. Before accepting a large shipment of steel wires, an engineer wants to be confident that µ is greater than 10, 000 pounds. (a) Identify the appropriate null
Problem 8.8 Suppose we use a sample of size n = 25 taken from a normal distribution with a 10 and the rejection region C = {T T c} to test the hypothesis Ho:15 against H1 : µ < 15. (a) Find the cutoff value c so that the test has significance level α = 0.1. Compute the probability of a type II
Problem 8.7 The thickness of metal wires used in the manufacture of silicon wafers is assumed to be normally distributed with mean µ (microns) and standard deviation σ = 1.0. To monitor the production process the mean thickness of 4 wires taken from each wafer is computed. The output is
Problem 8.6 Suppose we use a sample of size n = 9 taken from a normal distribution with σ = 4 and the rejection region C = {x : x < 28} to test the hypothesis H0: µ ≥ 30 against H1 : µ < 30. (a) What is the significance level of this test? Compute the probability of a type II error at the
Problem 8.5 The Environmental Protection Agency (EPA) requires that when the level of lead (measured in micro-grams per liter (µg/l)) reaches 15 µg/l in at least ten percent of the samples taken, then steps must be taken to reduce the levels of lead. Let Q(p) denote the pth quantile of the
Problem 8.4 The federal standard for cadmium dust in the workplace is 200µg/m3. To monitor the levels of cadmium dust an environmental engineer samples the air at 10 minute intervals, so each hourly average is the sample mean of 6 measurements. Assume that the level of cadmium dust is normally
Problem 8.3 The EPA standard for the MCL of arsenic in drinking water is 50 parts per billion (ppb). (a) Suppose one tests H0: µ ≤ 50 against H1 : µ > 50. Describe in this context the consequences of a type I and type II error. Which one has the more serious consequences? (b) Suppose one tests
Problem 8.2 Refer to the context of Problem 8.1. Suppose the water is regularly sampled for two weeks (so n = 14) and the observed radioactivity levels come from a normal distri[1]bution with unknown mean µ and known standard deviation σ = 0.4. The water is declared safe if x < 14.8; that is, we
Problem 8.1 The Environmental Protection Agency (EPA) standard for the maximum contaminant level (MCL) of alpha emitters in drinking water is 15 pCi/L (Picocuries per liter). Picocuries is a meaure of radioactivity. (a) Anecdotal evidence suggests that a town’s water supply fails to meet the EPA
Problem 7.60 It is sometimes necessary, e.g., in likelihood ratio tests, to compute the max[1]imum value of the likelihood function L(x|ˆθ). In particular, with reference to the likelihood function displayed in Equation 7.62 show that L(x|µ, ˆ σˆ2) = (ˆσ √ 2π) −ne−n/2.
Problem 7.59 Verify that µˆk, as defined in Equation 7.55 is an unbiased estimate for µk.
Problem 7.58 In a political campaign X = 185 voters out of n = 351 voters polled are found to favor candidate A. Use these data to compute the ML estimate of p and p(1 − p). Which of these estimators is unbiased? Discuss.
Problem 7.57 Let f(x|θ) = θxθ−1, 0(b) Use the result from part (a) to derive the method of moments estimator for θ.
Problem 7.56 The distribution of the lifetime of a component is assumed to have a gamma distribution with parameters α, β. The first two sample moments are µˆ1 = 96 and µˆ2 = 10368. Use the method of moments to estimate α, β.
Problem 7.55 The breaking strengths of 36 steel wires were measured and the following results were obtained: x = 9830 psi and s = 400. Assume that the measurements come from a normal distribution. The safe strength is defined to be the 10th percentile of the distribution. Compute the ML estimate of
Problem 7.54 Ten measurements of the concentration (in %) of an impurity in an ore were taken and the following data were obtained. 3.8 3.5 3.4 3.9 3.7 3.7 3.6 3.7 4.0 3.9 (a) Assuming these data come from a normal distribution N(µ, σ2) compute the ML esti[1]mators for µ, σ2, σ. (b) Which of
Problem 7.53(a) Let X1,...Xn be a random sample taken from an exponential distribu[1]tion with parameter λ. Show that X is an unbiased estimator of 1/λ. (b) Let U = min(X1,...,Xn). Show that U is exponentially distributed with parameter nλ. Hint: Show that P(U>x) = exp(−nλx). (c) Deduce from
Problem 7.52 Samples of size n1 = n2 = 150 were taken from the weekly output of two factories. The number of defective items produced were x1 = 10 and x2 = 15, respectively. Construct an approximate 95% confidence interval for p1 − p2, the difference in the propor[1]tion of defective items
Problem 7.51 To compare the effectiveness of two different methods of teaching arithmetic 60 students were divided into two groups of 30 students each. One group (the control group) was taught by traditional methods and the other group (the treatment group) by a new method. At the end of the term
Problem 7.50 Two machines produce computer memory chips. Of 500 chips produced by machine 1, 20 were defective; of 600 chips produced by machine 2, 40 were defective. Construct 90% one sided upper confidence intervals for: (a) p1, the proportion of defective chips produced by machine 1. (b) p2, the
Problem 7.49 How large a sample size is needed if we want to be 90% confident that the sample proportion will be within 0.02 of the true proportion of defective items and: (a) the true proportion p is unknown; (b) it is known that the true proportion p ≤ 0.1.
Problem 7.48 How large a sample size is needed if we want to be 95% confident that the sample proportion will be within 0.02 of the true proportion of defective items and: (a) the true proportion p is unknown; (b) it is known that the true proportion p ≤ 0.1.
Problem 7.47(a) To determine the level of popular support for a political candidate a newspaper regularly surveys 600 voters. The results of the poll are reported as a 95% confi[1]dence interval. Determine the margin of error. (b) Calculate the margin of error in part (a) when the sample size is
Problem 7.46 From a sample of 400 items 5% were found to be defective. Find a (a) 95% confidence interval for the proportion p of defective items. (b) 95% one sided upper confidence interval for p.
Problem 7.45 A random sample of 1300 voters from a large population produced the fol[1]lowing data: 675 in favor of proposition A and 625 opposed. Find a 95% confidence interval for the proportion p of voters: (a) in favor of proposition A. (b) opposed to proposition A.
Problem 7.44 From a random sample of 800 TV monitors that were tested, 10 were found to be defective; construct a 99% one sided upper confidence interval for the proportion p of defective monitors.
Problem 7.43 Let X1,...,Xn be a random sample from a Bernoulli distribution. Show that E(ˆp(1 − pˆ)) = n − 1 n × p(1 − p) < p(1 − p). Deduce that pˆ(1 − pˆ) is a biased estimator of p(1 − p).
Problem 7.42 Refer to the dopamine b-hydroxylase (DBH) activity data set of Problem 1.40. (a) Construct a 95% confidence interval for the difference in the DBH activity between the two groups of patients. (b) Construct a 95% confidence interval for the difference in the DBH activity between the two
Problem 7.41 The following data are from Charles Darwin’s study of cross- and self[1]fertilization. Pairs of seedlings of the same age, one produced by cross-fertilization and the other by self-fertilization, were grown together so that members of each pair were grown under nearly identical
Problem 7.40 Refer to the concentration of thiol in lysate data in Problem 1.62. (a) Derive a 95% confidence interval for the difference in the mean concentrations of thiol between the two groups. (b) Derive a 99% confidence interval for the difference in the mean concentrations of thiol between
Problem 7.39 With reference to the data set on lead levels in children’s blood (Problem 1.18) find a 100(1 − α)% confidence interval for the differences in lead level between the exposed and control group of children when: (a) α = 0.10. (b) α = 0.05. (c) α = 0.01.
Problem 7.38 To determine the difference, if any, between two brands of radial tires, 12 tires of each brand are tested and the following mean lifetimes and sample standard deviations for each of the two brands were obtained: x1 = 38500 and s1 = 3100 x2 = 41000 and s2 = 3500 Assume the tire
Problem 7.37 The skein strengths, measured in pounds, of two types of yarn are tested and the following results are reported. Yarn A 99 93 99 97 90 96 93 88 89 Yarn B 93 94 75 84 91 Construct a 95% confidence interval for the difference in means. Assume that both data sets come from a normal
Problem 7.36 An experiment was conducted to compare the electrical resistances (mea[1]sured in ohms) of resistors supplied by two manufacturers, denoted A and B, respectively. Six resistors from each manufacturer were randomly selected and their resistances measured; the following data were
Problem 7.35 Two different weight loss programs are being compared to determine their effectiveness. Ten men were assigned to each program, so n1 = n2 = 10, and their weight losses are recorded below. Construct a 95% confidence interval for the difference in weight loss between the two diets. Do
Problem 7.34 The compression strengths, measured in kgms, of cylinders produced by two different manufacturers were tested and the following data were recorded: X1 = 240, s1 = 10, n1 = 12 and X2 = 210, s2 = 7, n2 = 12. Construct a 95% confidence interval for the difference in mean compression
Problem 7.33 Two random samples of sizes n1 = 11 and n2 = 13, respectively, were taken from two normal distributions N(µ1, σ2) and N(µ2, σ2), and the following data were obtained: X1 = 1.5, s1 = 0.5, X2 = 2.1, s2 = 0.6. Find the values of: (a) s2 p. (b) s(X1 − X2). (c) a two sided 95%
Problem 7.32 A random sample of size n1 = 16 taken from a normal distribution N(µ1, 36) has a mean X1 = 30; another random sample of size n2 = 25 taken from a different normal distribution N(µ2, 9) has a mean X2 = 25. Find: (a) a 99% confidence interval for µ1 − µ2. (b) a 95% confidence
Problem 7.31 Derive Equation 7.33 for the 100(1 − α)% confidence value at risk.
Problem 7.30 An investor has $100,000 invested in the BEARX mutual fund. Assuming its weekly return and volatility are µ = −0.0046 and σ = 0.0290, respectively (see Table 1.13), find the 99% VaR (rounded to the nearest dollar) for: (a) a one week period. (b) a two week period.
Problem 7.29 An investor has $100,000 invested in the PTTRX mutual fund. Assuming its weekly return and volatility are µ = −0.0011 and σ = 0.0058, respectively (see Table 1.13), find the 99% VaR (rounded to the nearest dollar) for: (a) a one week period. (b) a two week period.
Problem 7.28 An investor has $100,000 invested in the S&P500. Assuming its weekly return and volatility are µ = 0.0031 and σ = 0.0255, respectively (see Table 1.13), find the 99% VaR (rounded to the nearest dollar) for: (a) a one week period. (b) a two week period.
Problem 7.27 A portfolio has a weekly return of µ = 0.003 and weekly volatility of σ = 0.06. The current value of the portfolio is $100,000. (a) Find the 99% VaR (rounded to the nearest dollar) for a one week period. (b) Find the 99% VaR (rounded to the nearest dollar) for a two week period.
Problem 7.26 The weekly mean return and volatility of the mutual fund BEARX for the year 1999 are −0.0046 and 0.0290, respectively (see Table 1.13). The sample size for these data is n = 51 (see Table 1.19). From these data find the mean return and volatility of the BEARX mutual fund for: (a) one
Problem 7.25 The weekly mean return and volatility of the bond mutual fund PTTRX for the year 1999 are −0.0011 and 0.0058, respectively (see Table 1.13). The sample size for these data is n = 51 (see Table 1.19). From these data estimate the mean return and volatility of the PTTRX mutual fund
Problem 7.24 The weekly mean return and volatility of the S&P500 index for the year 1999 are 0.0031 and 0.0255, respectively (see Table 1.13). The sample size for these data is n = 51 (see Table 1.19). From these data estimate the mean return and volatility of the S&P500 index for: (a) one year.
Problem 7.23 Refer to Example 7.8. The weekly mean return and volatility of GE for the year 1999 are 0.0091 and 0.0399, respectively (see Table 1.13). The sample size for these data is n = 51 (see Table 1.19). From these data estimate the mean return and volatility of GE’s stock for: (a) one
Problem 7.22 Refer to the data on the reverse-bias current in Problem 1.28. Assuming that the data come from a normal distribution, compute a 95% confidence interval for the variance.
Problem 7.21 Refer to the data set of Problem 1.19. Assuming the cadmium level is normally distributed, compute a 95% confidence interval for the variance of the cadmium levels.
Problem 7.20 Refer to the data set of Problem 1.18. Assuming each variable is normally distributed: (a) Compute a 95% confidence interval for the variance of levels of lead in the blood of the exposed group. (b) Compute a 95% confidence interval for the variance of levels of lead in the blood of
Problem 7.19 Refer to the data set of Problem 1.20. Assuming the capacitance is normally distributed, compute a 95% confidence interval for the variance of the capacitance.
Problem 7.18 Suppose the atomic weight of a molecule is measured with two different instruments, each of which is subject to random error. More precisely, the recorded weight Wi = µ+ei, where µ is the true atomic weight of the molecule and ei is the random variable that represents the error
Problem 7.17 Suppose the viscosity of a motor oil blend is normally distributed with σ = 0.2. (a) A researcher wants to estimate the mean viscosity with a margin of error not to exceed 0.1 and a 95% level of confidence; how large must n be? (b) Part (a) continued: Suppose he wants to estimate the
Problem 7.16(a) Compute a 99% confidence interval for the mean level of cadmium dust and oxide fume for the data set of Problem 1.19. Assume the data come from a normal distribution. (b) Compute a 95% upper confidence interval.
Problem 7.15(a) Compute a 95% confidence interval for the capacitances data in Problem 1.20. (b) Compute a 99% confidence interval.
Problem 7.14 An electronic parts factory produces resistors; statistical analysis of the output suggests that the resistances can be modeled by a normal distribution N(µ; σ2). The following data gives the results of the resistances, measured in ohms, for 10 resistors. Resistances: {0.150, 0.154,
Problem 7.13 Refer to the reverse-bias collector current of Problem 1.28. Assuming the data come from a normal distribution construct a 95% confidence interval for µ.
Problem 7.12 The lifetimes, measured in miles, of 100 tires are recorded and the following results are obtained: x = 38000 miles and s = 3200 miles. (a) If the data cannot be assumed to come from a normal distribution what can you say about the distribution of X? (b) Construct a lower one sided 99%
Problem 7.11 For a sample of 15 brand X automobiles the mean miles per gallon (mpg) was 29.3 with a sample standard deviation of 3.2; the gas tank capacity is 12 gallons. (a) Using these data determine a 95% one sided lower confidence interval for the mpg. Assume the data come from a normal
Problem 7.10 A sample of size n = 6 from a normal distribution with unknown variance produced s2 = 51.2 Construct a 90% confidence interval for σ2.
Problem 7.9 From a sample of size 30 taken from a normal distribution we obtain 1≤i≤30 xi = 675.2, 1≤i≤30 x2 i = 17422.5. (a) Find a 95% confidence interval for µ. (b) Find a 95% confidence interval for σ2.
Problem 7.8 Suppose that X1,...,X25 is a random sample from a normal population N(µ, σ2) and that x = 148.30. (a) Assume σ = 4; find a 95% confidence interval for µ. (b) Assuming s = 4, i.e., σ2 is unknown, obtain a 95% confidence interval for µ.
Problem 7.7 Eight measurements were made on the viscosity of a motor oil blend yield[1]ing the values x = 10.23, s2 = 0.64. Assuming the data come from a normal distribution N(µ, σ2) construct a 90% confidence interval for µ.
Problem 7.6 Let X1,...,Xn be a random sample from a normal distribution N(µ, 4), with µ unknown. Is ˆθ(X1,...,Xn) = 1≤i≤n (Xi − µ) 2 a statistic? Justify your answer.
Problem 7.5 (a) Let X1,...,Xn be a random sample from the Poisson distribution f(x; λ) = e−λ λx x! ; x = 0, 1,....Show that X is an unbiased estimator of λ. (b) Compute the standard error σ(X) and its estimated standard error s(X). (c) Refer to the data of Problem 1.72. Assuming the data
Problem 7.4 The price S(t) (t measured in years) of a stock is governed by a geometric Brownian motion with parameters µ = 0.12, volatility σ = 0.20, and initial price S0 = $75. Compute: (a) a 95% confidence interval for the stock price after 3 months. (b) a 95% confidence interval for the stock
Problem 7.3 Refer to the data set of Problem 1.18. The sample mean and sample standard deviation of the n = 33 measurements of the lead level in the blood of the control group of children are x = 15.88, s = 4.54. Assuming the data come from a normal distribution N(µ, σ2) compute: (a) a 90%
Problem 7.2 Refer to the data set of Problem 1.18. The sample mean and sample standard deviation of the n = 33 measurements of the lead level in the blood of the exposed group of children are x = 31.85, s = 14.41. Assuming the data come from a normal distribution N(µ, σ2) compute: (a) a 90%
Problem 7.1 A random sample of size n = 20 from a normal distribution N(µ, 16) has mean x = 75. Find: (a) a 90% confidence interval for µ. (b) a lower 90% confidence interval for µ. (c) a 95% confidence interval for µ. (d) a lower 95% confidence interval for µ.
Problem 6.20 Use Table A.6 in Appendix A to compute Fν1,ν2 (α) for: (a) F3,20(0.05); (b) F3,20(0.01); (c) F4,30(0.05); (d) F4,30(0.01).
Problem 6.19 A random sample of size 9 is taken from a normal distribution µ = 15 and σ unknown. Suppose s = 1.613. Find: (a) P(14 < X < 16); (b) P(13.76 < X < 16.24); (c) P(13.20 < X < 16.80).
Problem 6.18 Find the value c such that: (a) P(t14 ≥ c)=0.05; (b) P(|t14| ≥ c)=0.05; (c) P(t14 ≥ c)=0.01; (d) P(|t14| ≥c) = 0.01.
Problem 6.17 Use Table A.4 in Appendix A to compute tν(α) for: (a) t9(0.05); (b) t9(0.01); (c) t18(0.025); (d) t18(0.01).
Problem 6.16 The volatility of a stock is the standard deviation of its returns. Assume the weekly returns of a stock are normally distributed with σ = 0.042. Let s denote the sample standard deviation of the returns based on a sample of 13 weekly returns. (This corresponds to recording the
Problem 6.15 Suppose a random sample of size n = 16 is taken from a normal distribution with σ2 = 5. Compute the probability that the sample standard deviation s lies between 1.5 and 2.9.
Problem 6.14 The speed V of a molecule in a gas at equilibrium has pdf f(v) = a v2 e−b v2 , v ≥ 0; f(v)=0, otherwise. Here b = m/2kT , where m is the mass of the molecule, T is the absolute temperature, and k is Boltzmann’s constant. (a) Compute the constant a in terms ofb. (b) Let Y = (m/2)
Problem 6.13 An artillery shell is fired at a target. The distance (in meters) from the point of impact to the target is modeled by the random variable 7.22χ2 2. If the shell hits within 10 meters of its target it is destroyed. (a) Find the probability that the target is destroyed by the shell.
Problem 6.12 Use Table A.5 in Appendix A to compute numbers a b). (a) n = 10, p = 0.9 (b) n = 15, p = 0.9 (c) n = 10, p = 0.95 (d) n = 20, p = 0.95
Problem 6.11 Use Table A.5 in Appendix A to compute χ2 n(α) for: (a) n = 10, α = 0.1 (b) n = 10, α = 0.05 (c) n = 15, α = 0.1 (d) n = 20, α = 0.9
Problem 6.10 Boxes of cereal are filled by machine and the net weight of the filled box is a random variable with mean µ = 10 oz and variance σ2 = 0.5 oz2. A carton of cereal contains 48 boxes. Let T = total weight of carton. (a) Give a formula for the approximate distribution of T using the
Problem 6.9 An elevator has a weight capacity of 3000 lbs. Assume that the weight X of a randomly selected person is N(175, 400) distributed, i.e., µ = 175, σ = 20. Let Wn denote the total weight of n passengers. (a) Give the formulas for: E(Wn) and V (Wn). (b) Describe the distribution of Wn.
Problem 6.8 Assume that the miles per gallon (mpg) of two brands of gasoline come from the same normal distribution with σ = 2.0. Four cars are driven with brand A gasoline and six cars are driven with brand B gasoline. Denote the average mpg obtained with brand A and brand B gasolines by X and Y
Problem 6.7 To determine the difference, if any, between two brands of radial tires, 12 tires of each brand are tested. Assume that the lifetimes of both brands of tires come from the same normal distribution with σ = 3300. (a) Describe the distribution of the difference of the sample means X −
Problem 6.6 Let X denote the sample mean of a random sample of size n1 = 16 taken from a normal distribution N(µ, 36) and let Y denote the sample mean of a random sample of size n2 = 25 taken from a different normal distribution N(µ, 9). Assume X and Y are independent. (a) Describe the
Problem 6.5 Let X1,...,X9 be iid N(2, 4) distributed and Y1,...,Y4 be iid N(1, 1) dis[1]tributed. The Yi’s are assumed to be independent of the Xi’s. (a) Describe the distribution of X − Y . (b) Compute P(X > Y ).
Problem 6.4 Let X1,...,Xn be iid N(2, 4) distributed random variables. (a) If n = 100 compute P(1.9 < X < 2.1). (b) How large must n be so that P(1.9 < X < 2.1) = 0.9?
Problem 6.3 Suppose the time (measured in minutes) to repair a component is normally distributed with µ = 65, σ = 10. (a) What is the proportion of components that are repaired in less than one hour? (b) Give a formula for the distribution of the total time required to repair 8 components and use
Problem 6.2 A toy consists of three parts whose respective weights (measured in grams) are denoted by X1, X2, X3. Assume that the these weights are independent and normally distributed with X1 D = N(150, 36), X2 D = N(100, 25), X3 D = N(250, 60). (a) Give a formula for the distribution of total
Problem 6.1 Let X1, X2, X3 be three independent, identically distributed normal random variables with µ = 50, σ2 = 20. Let X = X1 − 2X2 + 2X3. Compute: (a) E(X). (b) V (X). (c) P(| X − 50 |≤ 25). (d) The 90th percentile of the distribution of X.
Problem 5.72 Let X and Y have a bivariate normal distribution with parameters µX = 2, µY = 1, σ2 X = 9, σ2 Y = 16, ρ = −2/3. Compute: (a) P(X < 4) (b) P(X < 4|Y = 1) (c) E(X|Y = 1) (d) V (X + Y )
Problem 5.71 Let X and Y have a bivariate normal distribution with parameters µX = 2, µY = 1, σ2 X = 9, σ2 Y = 16, ρ = −3/4. Compute: (a) P(Y < 3) (b) P(Y < 3|X = 2) (c) E(Y |X = 2) (d) V (X + Y )
Problem 5.70 Let X and Y have a bivariate normal distribution with parameters µX = 2, µY = 1, σ2 X = 9, σ2 Y = 16, ρ = 3/4. Compute: (a) P(Y < 1) (b) P(Y < 1|X = 0) (c) E(Y |X = 0) (d) V (X + Y )
Problem 5.69 Let X and Y have a bivariate normal distribution with parameters µX = 2, µY = 1, σ2 X = 9, σ2 Y = 16, ρ = −3/4. Compute: (a) P(Y < 3) (b) P(Y < 3|X = 2) (c) E(Y |X = 2) (d) V (X + Y )
Problem 5.67 Let the random vector (X, Y ) have the joint density function f(x, y) = xe−xy−x, for x > 0,y> 0) f(x, y)=0, elsewhere. Compute: (a) f(y|x). (b) µY |x. Problem 5.68 Let X and Y have a bivariate normal distribution with parameters µX = 2, µY = 1, σ2 X = 9, σ2 Y = 16, ρ = 3/4.
Problem 5.66 Let X, Y have joint density function given by f(x, y) = 3 5 xy + y2 , 0 ≤ x ≤ 2, 0 ≤ y ≤ 1 f(x, y)=0, elsewhere. Compute: (a) P(Y < 1/2|X < 1/2). (b) f(y|x). (c) E(Y |X = x).
Problem 5.65 Let (X, Y ) be uniformly distributed over the unit circle {(x, y) : x2 + y2 ≤ 1}. Its joint distribution function is given in Problem 5.62. Find: (a) the marginal distributions. (b) Show that Cov(X, Y )=0, but that X and Y are not independent.
Problem 5.64 Let X, Y have the joint pdf f(x, y) = 6 7 (x + y)2, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, f(x, y)=0,elsewhere. By integrating over an appropriate region compute the following probabilities: (a) P(X + Y ≤ 1). (b) P(Y >X). (c) Compute the marginal pdfs: fX(x) and fY (y).
Problem 5.63 Let X, Y have the joint pdf f(x, y) = c(x + xy), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, f(x, y)=0,elsewhere. (a) Compute the value ofc. (b) Compute the marginal pdfs fX(x) and fY (y). (c) Are the random variables X and Y independent? (d) Compute P(X + Y ≤ 1) by first writing it as a double

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