I'm taking a probability theory class and I'm having troubles with multivariate distributions. In particular, I don't really understand how to find conditional probabilities.
Here's a question I'm struggling with: If f(y1, y2)=1 for y1 and y2 between 0 and 1, and 0 elsewhere, Find P(0.3 less than or equal to Y1 less than or equal to 0.5 given that Y2=0.3).
I understand intuitively that the answer is 0.2, but I don't really get how to solve the equation and my textbook isn't very clear. Any help would be greatly appreciated! I can't really solve any of the harder question because I don't seem to understand this basic concept.
You have to calculate $P(0.3 \leq Y_1 \leq 0.5|Y_2 \leq 0.3)$. It is $y_2=0.3$
And the marginal densitiy function of $Y_2$ with the limit $y_2=0.3$ is
$f_{Y2}(y_2)=\int_0^{y_2} f(y_1,y_2)\ dy_1=\int_0^{0.3} f(y_1,y_2)\ dy_1=0.3$
Thus the conditional probability is $P(0.3 \leq Y_1 \leq 0.5|Y_2 \leq 0.3)=\frac{\int_{0.3}^{0.5} \int_0^{0.3} f(y_1,y_2) \ dy_2 \ dy_1}{\int_0^{0.3}f(y_1,y_2) \ d{y_1}}=\frac{\int_{0.3}^{0.5} \int_0^{0.3} 1 \ dy_2 \ dy_1}{\int_0^{0.3}1 \ d{y_1}}$