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The article “Microwave Observations of Daily Antarctic Sea-Ice Edge Expansion and Contribution Rates” (IEEE Geosci. and Remote Sensing Letters, 2006: 54–58) states that “The distribution of the daily sea-ice advance/retreat from each sensor is similar and is approximately double exponential.” The proposed double exponential distribution has density
a. What is the value of the parameter λ?
b. What is the
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Probability and Statistics for Engineering and the Sciences
- Let the continuous random variable X denote the current measured in a thin copper wire in milliamperes. Assume that the range of X is [4.9, 5.1] mA, and assume that the probability density function of X is f(x) = 5 for 4.9 <= x <= 5.1. What is the variance?arrow_forward1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.arrow_forwardAssume that the range of X is [4.9, 5.11 mA, and assume that the probability density function of X is f(x) = 5 for 4.9 <= x <= 5.1. What is the variance?arrow_forward
- The article “Monte Carlo Simulation—Tool for Better Understanding of LRFD” (J. of Structural Engr., 1993:1586–1599) suggests that yield strength (ksi) for A36 grade steel is normally distributed with a mean of 50.2 ksi and a standard deviation of 3.8 ksi. What is the probability that the yield strength is at most 45? (Use 4 decimal places). Show the area of interest in the density curve.arrow_forwardFor a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)arrow_forwardConsider the following 10 measurements about the cohesion of a soil: 12 kPa, 14 kPa, 15 kPa, 14.5 kPa, 16.0 kPa, 18 kPa, 15 kPa, 16.2 kPa, 17. 6 kPa, 13 kPa. Work out the solutions of the following two problems, assuming the cohesion follows the normal and lognormal distributions, respectively. (1) Draw the PDF of the cohesion; (2) Evaluate the probability that the cohesion based on the PDF is less than 5 kPa, 10 kPa, and 15 kPa; (3) Assess the effect of the type of distribution on the results.arrow_forward
- Suppose the duration (in months) of a construction job can be modeled as a continuous random variable T whose cumulative distribution function (CDF) is given by??(?) = ?2 − 2? + 1 1 ≤ ? ≤ 2??(?) = 0 ? < 1??(?) = 1 ? > 2a. Determine the corresponding density function, ??(?).b. Compute ?(? > 1.5).arrow_forwardFor a laboratory assignment, if the equipment is working, the density function of the observed outcome X is f(x) = 2x(1-x), 0 < x < 1 Find the variance and standard deviation of Xarrow_forwardGiven the sample 0.1, 0.4, 0.8, 0.8, 0.9, you wish to test the goodness of fit of the distribution with a probability density function given by: f(x)= (1+2x)/2, 0 ≤ x ≤ 1 Determine the Kolmogorov-Smirnov goodness of fit statistic.arrow_forward
- A type of network router has a bandwidth total to first hardware failure called ? expressed in terabytes. The random variable ? is modeled by a distribution whose density is given by one of the following functions: fs(s) = 1/? for s [0,?] with a single parameter ?. Consider the bandwidth total to failure ? of the sequence of the two routers of the same type (one being brought up automatically when the first is broken).Express ? in terms of the bandwidth total to failure of single routers ?1 and ?2. Formulate realistic assumptions about these random variables. Calculate the density function of the variable ?. Given an experiment with the dual-router-system yielding a sample ?1 , ?2 , ..., ?? , calculate the likelihood function for ?. Propose a transformation of this likelihood function whose maximum is the same and can be computed easily. An actual experiment is performed, the infrastructure team has obtained the bandwidth totals to failure given by the sequence ?10 of numbers (225, 22,…arrow_forwardThe article “Monte Carlo Simulation—Tool for Better Understanding of LRFD” (J. of Structural Engr., 1993:1586–1599) suggests that yield strength (ksi) for A36 grade steel is normally distributed with a mean of 50.2 ksi and a standard deviation of 3.8 ksi. What is the probability that the yield strength is greater than 60? (Use 4 decimal places) Show the area of interest in the density curve.arrow_forwarda. What is the probability that the lifetime X of the first component exceeds 3? b. What are the marginal pdf's of X and Y? Are the two lifetimes independent? x. What is the probability that the lifetime of at least one component exceeds 3?arrow_forward
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