Question

Suppose that the random variables  X  and Y  have the following joint probability density function.

f (x, y)  =  ce^{−5x − 7y},    0  <  y  <  x.

(a)	Find the value of c.

(b)	Find P(X  < 1/3  , Y  <  2)

Answer 1

a)

for this to be valid:

f(x,y) dy dx must be 1

f(x,y) dy dx =c*(e^-5x-7y) dy dx =c*e^-5x*(-e^-7y/7)|^x₀ dx

= (c/7)e^-5x(1-e^-7x) dx =(c/7)*(-e^-5x/5+e^-12x/12) |₀ =(c/7)*(1/5-1/12)=(c/7)*(7/60)=1

c =60

b)

P(X<1/3,Y<2)=f(x,y) dy dx = (60/7)*(-e^-5x/5+e^-12x/12) |^1/3₀ =0.689296