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- Suppose that X, Y are jointly continuous with joint probability density function f (x, y) = ce ² (² 2 x, y = (-∞0, ∞0), for some constant c. (a) Find the value of the constant c. (b) Find the marginal density functions of X and Y.In physiology, an objective measure of aerobic fitness is how efficiently the body can absorb and use oxygen (oxygen consumption). A physiologist, Dr. Castillejo, conducted a research wherein subjects participated in a predetermined exercise run of 1.5 miles. Measurements of oxygen consumption as well as several other variables such as age, gender, runtime, resting and maximum pulse rates, and weight were recorded from 50 randomly selected gym members. Dr. Castillejo is interested in determining whether any of these other variables can help predict oxygen consumption. She believes that a possible link between these factors will help her determine how to improve the health of her gym members. Help Dr. Castillejo in developing a proposal for the improvement of her gym members' health condition. The variables are described below. gender- either male or female runtime-time to run 1.5 miles (in min) age age of the gym member in years weight weight of the gym member (in kg) oxygen…The data for the joint probability mass function of X and Y (two different measurement systems) aregiven in the table below.a) Calculate the marginal distributions of X and Y and plot them.b) Select one of the Y values from the table and find the conditional probability mass function of X for thatY value you have selected and plot it.c) Show whether X and Y are independent or not.d) Calculate the covariance of (X,Y) i.e. Cov(X,Y).
- The cumulative distribution function of the continuous uniform distribution between con- stants a and b is given by F(x) = P(X ≤ x) = x-a - a x b (a) The probability density function is f(x) = F(r). Find the form of f(x). (b) Find the derivative of f(x) for x = [a, b]. Is f(x) decreasing, increasing or flat in this region? (c) Does f(x) have a single maximum in the region [a,b]? If so, what is it, or if not, why not?2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 02. The continuous random variables X and Y have known joint probability density function f(x,y) given by £xy (x, y) = {(x + y) [(x+y)/8 0 ≤x≤ 2, 0≤y≤ 2 fxx otherwise Define the random variable z as Z = min(X,Y). a) Determine the probability density function fz(z) of the random variable z. b) Determine the probability Pr{0.5Suppose that X and Y have a joint probability density 7e-2-7y if x, y ≥ 0 otherwise (a) Verify that fx.y is indeed a probability density function. function given by fx.x (x, y)Driving test You are about to take the road test for yourdriver’s license. You hear that only 34% of candidatespass the test the first time, but the percentage rises to72% on subsequent retests.a) Create a plan for a simulation to estimate theaverage number of tests drivers take in order to geta license.b) Here are results of 100 trials of a simulation. Use theseresults to estimate the average number of tests driverstake in order to get a license.1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.Let X and Y be jointly continuous with joint probability density function f(x, y) and marginal densities fX(x) and fY(y). Suppose that f(x, y) = g(x)h(y) where g(x) is a function of x alone, h(y) is a function of y alone, and both g(x) and h(y) are nonnegative. a) Show that there exists a positive constant c such that fX(x) = cg(x) and fY(y) = (1/c)h(y). b) Use part (a) to show that X and Y are independent.Let X and Y have the joint probability density function fxy(x, y) = {#², for 0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. FreemanMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman