9. Let X be a random variable with density function f and characteristic function . Show, subject to an appropriate condition on f, that 1 [ f(x)² dx = = = 100 16 (1)1² dt. 2π 8
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- 2. Let X be a continuous random variable with probability density function (pdf) 3x�� 0Let 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.5. Let 0 and a be parameters. Show that the family of functions fo,a(x) = I > 0 (x+0)a+1³ is a probability density function. What can you say about the mean of a random variable that has fo,a as a pdf?Consider the following density function 4z(9–z²) 81 05. Suppose that two continuous random variables X and Y have joint probability density function (e**y +e 2x-y) 15xs2,0sys3 elsewhere a. P( 3/2 s Xs2, 1SYS2) b. Are the random variables X and Y independent? c. find the conditional density X given Y = 0 d. Find the strength of the relationship and interpret the findings.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 04. Let 0 be a parameter. Show that the family of functions Oe-0/x fo(x) = I > 0 is a probability density function. What can you say about the mean of a random variable that has fo as a pdf?2. Let X be a Chi-squared random variable with density function fx(x) = exp x > 0. 2 Suppose Y is independent of, and has the same distribution as, X.5. Given a normalized probability density function P(x) of finding the variable x in the interval [x,x + dx], write the definition for a) the mean value (x), b) the variance o? = ((x – (x))?), and c) the standard deviation of the variable x.1. Let X1,..., X, be a random sample from a distribution with pdf f(x) = 2x, 0< x < 1. Compute the joint density of the smallest and the largest order statisics.6.74 Let Y₁, ₂, ..., Y, be independent, uniformly distributed random variables on the interval [0, 0]. Find the a probability distribution function of Y() = max(Y₁; Y₂;; Y₁). b density function of y(n). c mean and variance of y(n).Suppose that the random variable W has a beta distribution with probability density function f (w) given as f(w) = Kwi(1 – w),0 < w < 1 Find the value of K and hence the average value of W.SEE MORE QUESTIONSRecommended textbooks for youGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning