a)for above to be valid: f(x,y) dy dx =1
f(x,y) dy dx =kx(1+3y2) dy dx = kx(y+y3)|10 dx =kx(1+1) dx =2kx2/2|20 =4k =1
k=1/4
g(x) =f(x,y) dy=x(1+3y2)/4 dy =(x/4)*(y+y3)|10 =x/2
h(y) =f(x,y) dx=x(1+3y2)/4 dx =(x2/8)*(1+3y2)|20 =(1/2)*(1+3y2)
f(x|Y) =f(x,y)/h(y) =(1/4)x(1+3y2)/(1/2)*(1+3y2)) =x/2
b)P(1/4<X<1/2)= f(x) dx =x/2 dx = x2/4|1/21/4 =1/16-1/64 =3/64
7. Given the joint density function /(x,y) =(kx (1 + 3 y*) 0<x<2,0<p?1 elsewhere a. Find...
1. The density function of b is given by kx(1 - x) f(x) = { for 0 < x 51, elsewhere. (a) Find k and graph the density function. (b) Find P(1/4 < ſ < 1/2). (c) Find P(-1/2 sã < 1/4). (d) Find the CDF and graph it. (e) Find E( ), E(52), and V(5). 1. The density function of ğ is given by |kx(1 – x) o for 0 < x 51, elsewhere. f(x (a) Find k and...
2) Suppose that X has density function f(a)- 0, elsewhere Find P(X < .3|X .7).
The probability density function of X is given by 0 elsewhere Find the probability density function of Y = X3 f(r)-(62(1-x)for0 < x < 1
Assume that the joint density function of X and Y is given by f (x, y) = 4,0 < x < 2,0 < y = 2 and f (x, y) = 0 elsewhere. (a) Find P (X < 1, Y > 1). (b) Find the joint cumulative distribution function F(x, y) of the two random variables. Include all the regions. (c) Find P (X<Y). (d) Explain how the value of P (1 < X < 2,1 < Y < 2)...
Let the joint density function of random variables X and Y be f(x,y) = 8 - x - y) for 0 < x < 2, 2 < y < 4 0 elsewhere Find : (1) P(X + Y <3) (11) P(Y<3 | X>1) (111) Var(Y | x = 1)
[2.5 points] If two random variables have a joint density given by, f(x, y) = k(3x + 2y) 0 for 0 < x < 2, 0 < y < 1 elsewhere (a) Find k (b) Find the Marginal density of Y. (c) Find E(Y) (d) Find marginal density X. (e) Find the probability, P(X < 1.3). (f) Evaluate fı(x|y); (g) Evaluate fi(x|(0.75))
7. Suppose that the joint density of X and Y is given by f(x,y) = e-ney, if 0 < x < f(z, y) = otherwise. Find P(X > 1|Y = y)
(8pts) 1. The joint probability density of X and Y is given by + 0<x<1 and 0 <y< 2 otherwise a) Verify that this is a joint probability density function. b) Find P(x >Y). o) Find Pſy > for< d) Find Cov(X,Y). e) Find the correlation coefficient of X and Y (Pxy).
# 6 If two random variables have the joint density f(x, y)=59 y?) for 0<x<1, 0<y<1 0 elsewhere a. Find the probability that 0.2 X<0.5 and 0.4<Y<0.6. b. Find the probability distribution function F(x, y). c. Are x and y independent?
7. Show that if the joint probability density function of X and Y is if 0 < x <.. =sin(x + y) f(x, y) = { VI fres 9 Line + »» Hosszž, osys elsewhere, then there exists no linear relation between X and Y.