1. Let X be the amount of water in a two gallon container in the fridge of the Jones household at the start of the day and let Y denote the amount dispensed rom that two gallon container during that day. We assume that (X,Y) has joint density. f(x,y) = (xy)/2, 0<y<x<2.
a. Find the marginal density of X.
b. Find E(XY)
c. find the conditional density off Y given X=x
d. Are X and Y independent?
1. Let X be the amount of water in a two gallon container in the fridge...
3. Let X denote the temperature (°C) and let Y denote the time in minutes that it takes for the diesel engine on an automobile to get ready to start. Assume that the joint density for (X,Y) is given by fxy(x, y) = c(4x + 2y + 1),0 < x < 40,0 < y = 2 (a) Find the value of c that makes this joint density legitimate. (b) Find the probability that on a randomly selected day the air...
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
Let X, Y be jointly continuous with joint density function (pdf) fx,y(x, y) *(1+xy) 05 x <1,0 <2 0 otherwise (a) Find the marginal density functions (pdf) fx and fy. (b) Are X and Y independent? Why or why not?
Let f(x, y) = ( kxy + 1 2 if x, y ∈ [0, 1] 0 else denote the joint density of X and Y a) Find k b) Find the marginal density of X (because of the symmetry of the joint pdf, the marginal density of Y is analogous). c) Determine whether X and Y are independent. d) Find the mean of X e) Find the cumulative distribution function of X. Set up an equation (but no need to...
1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a. Find the joint pdf fxy(x, y), marginal pdf (fx(x) and fy()) and cdf (Fx(x) and Fy)) b. Find the conditional pdf fxiy Cr ly c. Find the probability P(X < Y = y) d. Are X and Y independent?
4. Let X denote the number eggs hatched out of Y eggs laid by a particular parasite. The joint pmf of (X, Y) is given by A(1-0) e ,for x 0, 1, 2,.., y , and y = 0,1,2, .,00 Px,y (x, y)= 1 -6 = 0, otherwise where A> 0 and 0< 0<1 are unknown constants. (a) Find the marginal pmfs of X and Y. Are X and Y independent? (b) Find the conditional pmf of X|Y = y...
2. Let the joint pdf of X and Y be given by f(xy)-cx if 0sysxsi Determine that value of c that makes f into a valid pdf. a. Find Pr(r ) b 2 C. Find Prl X d. Find the marginal pdf's of X and Y e. Find the conditional pdfs of 자리 and ri- f. Are X and Y independent? Give a reason for your answer g. Find E(X), E(Y), and E(X.Y)
2. Let the joint pdf of X...
Two balls are placed randomly into two boxes labeled as I and II. Let X denote the number of balls in box I and Y denote the number of occupied boxes. (a) Find the joint density function of X and Y. (b) Compute E(X) and the conditional expectation E(X|Y= 1).