Y₁ and Y₂ are independent, so their joint density is
[tex]f_{Y_1,Y_2}(y_1,y_2)=f_{Y_1}(y_1)f_{Y_2}(y_2)=\begin{cases}\frac1{49}e^{-\frac{y_1+y_2}7}&\text{for }y_1\ge0,y_2\ge0\\0&\text{otherwise}\end{cases}[/tex]
By definition of conditional probability,
P(Y₁ > Y₂ | Y₁ < 2 Y₂) = P((Y₁ > Y₂) and (Y₁ < 2 Y₂)) / P(Y₁ < 2 Y₂)
Use the joint density to compute the component probabilities:
• numerator:
[tex]P((Y_1>Y_2)\text{ and }(Y_1<2Y_2))=\displaystyle\int_0^\infty\int_{\frac{y_1}2}^{y_1}f_{Y_1,Y_2}(y_1,y_2)\,\mathrm dy_2\,\mathrm dy_1[/tex]
[tex]=\displaystyle\frac1{49}\int_0^\infty\int_{\frac{y_1}2}^{y_1}e^{-\frac{y_1+y_2}7}\,\mathrm dy_2\,\mathrm dy_1[/tex]
[tex]=\displaystyle-\frac17\int_0^\infty\int_{-\frac{3y_1}{14}}^{-\frac{2y_1}7}e^u\,\mathrm du\,\mathrm dy_1[/tex]
[tex]=\displaystyle-\frac17\int_0^\infty\left(e^{-\frac{2y_1}7} - e^{-\frac{3y_1}{14}}\right)\,\mathrm dy_1[/tex]
[tex]=\displaystyle-\frac17\left(-\frac72e^{-\frac{2y_1}7} + \frac{14}3 e^{-\frac{3y_1}{14}}\right)\bigg|_0^\infty[/tex]
[tex]=\displaystyle-\frac17\left(\frac72 - \frac{14}3\right)=\frac16[/tex]
• denominator:
[tex]P(Y_1<2Y_2)=\displaystyle\int_0^\infty\int_{\frac{y_1}2}^\infty f_{Y_1,Y_2}(y_1,y_2)\,\mathrm dy_2\,\mathrm dy_1=\frac23[/tex]
(I leave the details of the second integral to you)
Then you should end up with
P(Y₁ > Y₂ | Y₁ < 2 Y₂) = (1/6) / (2/3) = 1/4
