Probability- Transformation of variable

[jack1234]

Following is the
question
http://tinyurl.com/62uxof
solution
http://tinyurl.com/6m4lcc

The distribution in question means cumulative distribution.

What I do not understand in the solution is the step from P{X_1<=wX_2} to the integration formula that followed immediately. May I know the reasoning for it?

 

[tiny-tim]

What I do not understand in the solution is the step from P{X_1<=wX_2} to the integration formula that followed immediately. May I know the reasoning for it?

Hi jack1234!

P(X ≤ wY) = lim ∑P(X ≤ wY | v ≤ Y ≤ v + dv)P(v ≤ Y ≤ v + dv)

= ∫P(X ≤ wv)dP(Y ≤ v)