2D Conditional Probability

#1
Question
X, Y are two independent random variables with cumulative distribution functions:

F_X(x) =
{0, x<0;
x/3, 0<x<3;
1, x > 3}

F_Y(y) =
{0, y<0;
y^2/16, 0<y<4;
1, y>4}

Find P(X<2, Y<1| X<1, Y<3)

Rationale
I figured that P(X<2|X<1) = 1 because if X<1, then it follows that X<2. But I am unsure how to calculate this overall probability. I think having two dimensions to the problem is tripping me up. Further, I figured that finding P(Y<1|Y<3) should give me an answer since X and Y are independent I should be able to multiply this probability by 1 for the final answer, but I am unsure how to calculate this. Any help would be appreciated! Thanks!
 
Last edited:

BGM

TS Contributor
#2
1. Use the definition of conditional probability to simplify the expression.

2. Use the given CDF to calculate the probabilities.
 
#3
Okay,

So P(X<2, Y<1|X<1, Y<3) = P(X<2|X<1)*P(Y<1|Y<3) by independence. The first term equals 1 from the reasoning in the original post. Would the second term simplify to P(Y<1)/P(Y<3)? I don't have a formal reasoning to get to this, but intuitively, that would seem to fall out.