Suppose X,Y have joint density function f(x,y) = (12/7)(xy+y^2), 0 <= x <= 1 and 0 <= y <= 1 and 0, otherwise. a) Check that f is a genuine joint density function. b) Find the marginal density functions of X and Y. c) Calculate the probability P(X < Y).
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Suppose X,Y have joint density function
f(x,y) = (12/7)(xy+y^2), 0 <= x <= 1 and 0 <= y <= 1 and 0, otherwise.
a) Check that f is a genuine joint density function.
b) Find the marginal density
c) Calculate the probability P(X < Y).
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- A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in inches. a. Using the fact that the volume of the can is 25 cubic inches, express h in terms of x. b. Express the total surface area S of the can in terms of x.The joint density of X and Y is given by, ху, Iw(x, y) = (x² +: 0 l) b) Find the marginal probability distributions of X and Y. c) Find the conditional probability density function of Y given X=0.5 and calculate P(YA student has 3 volumes of short stories and 2 novels on a bookshelf. She selects 3 books at random to take home over vacation. A random variable X is defined to be the number of novels selected. Find the values assumed by X and its density function.fxlx,y) = x2y is a joint probability density function over the range 0The joint density function of Y1 and Y2 is given byf(y1, y2) = (y1 + y2, 0 < y1 < 1, 0 < y2 < 1,0, elsewherea) Find marginal density functions. Are Y1 and Y2 independent?b) Could you tell whether Y1 and Y2 were independent or not without finding marginaldensity functions? Explain.c) Find P(Y1 + Y2 < 0.2).d) Find P(0 < Y2 < 0.5|Y1 = 0.4).The joint probability density function of two dimensional random variable(X,Y) is given 8 by f(x, y) = xy, 13xsys2 9 = 0, elsewhere Find the marginal density functions of X and Y.There are two components in an electronic system. Let X and Y denote the lengths of life in years. The joint density function of these variables f(x, y) = {ye-&+x) ,x 2 0, y Determine P(x>3) What is the probability that both components will exceed 3 years?Let X denote the amount of space occupied by an article placed in a 1-ft3 packing container. The pdf of X is f(x) = A.x^a(1 - x), 0 a) Show that it is complete probability density function. b) What is P(X < lambda / lambda + gamma) c) Compute mean and variance. d) Find the distribution function.Let X and Y have the following joint probability density function: ficylx,y) = 1 / (x² y²), for-1, y>1 0, otherwise Let U = 6 XY and V = 6 X. In all question parts below, give your answers to three decimal places (where appropriate). (a) The non-zero part of the joint probability density function of U and V is given by fuylu,v) = A v® uC for some constants A, B, C. Find the value of A. Find the value of B. (b) The support of (U,V), namely the values of u and v that correspond to the non-zero part of fyylu,v) given in part (a), is given by: O u>1.00 and v>6.00 O u> 6.00 and v6.00 and v> 6.00 O u>1.00 v and v>6.00 O None of these answers is correctQ1) Discrete joint variables X and Y with probability density f(x,y) (pdf) are given in this table. Find: 1) The Covariance Cov(X,Y)? (Cov(X,Y) = MxY-MxMY) 2) The correlation (pxy) between X and Y where Pxy = Cov(X,Y) PXPY Y 3 fx(x) f(x,y) 1 2 1 1/4 1/4 0 X 2 0 1/4 1/4 fy(y) Note that 2 n=2 n=3 Px² = Σn²±²x² f(x, y) - μ and py² = Σ3y² f(x, y) – µ Zk=06. Determine the value of c that makes the function f(x,y) = cxy a joint probability %3D density function over the range 01) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.SEE MORE QUESTIONSRecommended textbooks for youFunctions and Change: A Modeling Approach to Coll…AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll…AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning