Question

AB and CD are two lines of length 10 cm. and 5 cm. respectively.

We choose point P on line AB and point Q on line CD at random.

Length of CQ=X and length of AP=Y are two independent random variable

a)   P(Y > X)=?

b)  P( X=2. Y=2)=?

Answer 1

P_X(x)=1/5 for 0$\le X \le 5$

P_Y(y)=1/10 for 0$\le Y \le 10$

P(Y > X)=$\int_{x=0}^{5} \int_{y=x}^{10} P_X(x)*P_Y(y) dydx= \int_{x=0}^{5} \int_{y=x}^{10} 1/50 dydx=$

$\int_{x=0}^{5} (10-x)/50*dx=-1/50*\int_0^5x*dx+1/5\int_0^51*dx=-5^2/100+5/5=3/4$

P( X=2 and Y=2) = P(X=2)*P(Y=2) = $(\int_2^{2} 1/5*dx)*(\int_2^{2} 1/10*dy)=0$