Question

I

Answer 1

(a)

P(1/2 X 3/2) = F(3/2) - F(1/2)

= (1/2) * (3/2) - (1/2) * (1/2) {F(x) = x/2 for 0 x 2}

= (3/4) - (1/4)

= 1/2

(b)

P(1 X < 2) = F(2) - F(1)

= (1/2) * 2 - (1/2) * 1 {F(x) = x/2 for 0 x 2}

= 1 - (1/2)

= 1/2

(c)

Y X

=> X² X

=> X - X² 0

=> X (1 - X) 0

which is true if X 0 and X 1 (or 0 X 1)

or X 0 and X 1 (which is not possible at the same time.)

Thus,

P(Y X ) = P(0 X 1) = F(1) - F(0)

=  (1/2) * 1 - (1/2) * 0

= 1/2

d)

X 2Y

=> X 2X²

=> 2X² - X 0

=> X (2X-1) 0

which is true if X 0 and X 1/2 (=> X ​ 1/2)

or X 0 and X 1/2 (=> X 0)

Thus,

P(X 2Y ) = P(X ​ 1/2) + P(X 0) = 1 - P(X 1/2) + 0 {F(X) = 0 for X < 0}

= 1 - F(1/2)

= 1 - (1/2) * (1/2)

= 1 - (1/4)

= 3/4

(e)

X + Y 3/4

=> X + X² 3/4

=> X² + X - (3/4)   0

=> X² + X - (3/4)   0

=> (X + 3/2) (X - 1/2) 0

which is true if X -3/2 and X 1/2 (which is not possible at the same time.)

or X -3/2 and X 1/2    (or -3/2 X 1/2)

Thus,

P(X + Y 3/4 ) = P(-3/2 X 1/2) = F(1/2) - F(-3/2)

= (1/2) * (1/2) - 0 {F(x) = 0 for X < 0 and F(x) = x/2 for 0 x 2)

= 1/4