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- A density curve consists of the line segment connecting the points (0,1) and (0.5,1) and the segment connecting(0.5, 1) to the x-axis.a. Determine the coordinate point where the second segment crosses the x-axis.b. Determine the slope of that segmentc. Determine the equation of the line containing this segment (y = mx + b)d. Calculate the probability P(X > 1)3. Joint distribution for continuous X, Y. Let X1, X2 denote two independent exponential r.v.'s with X; ~ Exp(1). Let X1 %3D X1 + X2 Find fy(y). Hint: recall the pdf for an exponential r.v., f(x) = de-x.Let the density function of a random variable X be f (x) = xe¯* for x > 0. Find The moment generating function, P(X = 1) and P(X = 2). b. The mean and the variance of the distribution of X. а.
- 3. The storm runoff X (in cubie meters per second, ems) can be modeled by a random variable with the following probability density function (PDF): fa(x) = c (x -) for 0Suppose that Y has a gamma distribution with parameters a and . a) If d is any positive or negative value such that a+d>0, show that ed r(a+d) E (Y*) – T(a) b) Use the result in part (a) to give an expression for E ( /1/Y) . What do you need to assume about a ? c) Show that, with d = 1, the result in part (a) gives E(Y) = a05. Let Z be a standard normal random variable. Obtain moment generating function (mgf) of Z.Find the density of random variable Y = Z2 and verify that it is a pdf.3.6 The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function f(x): = 20,000 (x+100)³, 0, x > 0, elsewhere. Find the probability that a bottle of this medicine will have a shell life of (a) at least 200 days; (b) anywhere from 80 to 120 days.Determine whether the function is a probability density function over the given interval. 35) f(x)= x, [0, 8] 32 A) No B) Yes mohabiliter domait. C. e interval find the mean the variance For thosEstimate θ by using method of moment.A5)rv X and Y have a joint Gaussian distribution ~ N(µ =[0.5,1.6], Σ a. Where is the centroid of the distribution, i.e., (X,Y) coordinates? b. Are X and Y independent? Explain why or why not = C. What is the standard deviation of X, σx? [0.5 0.2)] 10.2 0.3. d. In the covariance matrix, if 0.5 increased to 1.5, what would happen to the distribution? e. What is cov(X,Y)? f. Sketch the approximate shape of this distributionLet X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article, "Blade Fatigue Life AsSsessment with Applications to VAWTS," J. Solar Energy Engr., (1982), pp 107--111, proposes the Rayleigh distribution as a model for the X distribution. Here is the pdf of the model. i e/(2 0), x> 0, f(x, 0) = 0, otherwise. 1 1.1. Suppose that 0= 100 (a value suggested by a graph in the article). What is the probability that X is at most 200? Less that 200? At least 200? 1.2. What is the probability that X is between 100 and 200, assuming 0 = 100, as before? 1 1.3. Determine the expression of the cdf, F(x, 0) = P(XSEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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