Thread: Probability questions. Please kindly help.

1. Let Y1 and Y2 be independent random variables that are both uniformly distributed on the interval (0,1). Find P(Y1< 2*Y2|Y1< 3*Y2).

2. Let Z ~ N(0,1) and Y ~ χ2(ν) (Chi-square with ν degrees of freedom). Assume that Z is independent of Y and let W = Z/√Y . Obtain E(W) and Var(W).

3. The joint density of Y1 and Y2 is given by
f(y1,y2) = {y1+ y2, 0≤ y1≤ 1, 0 ≤ y2≤ 10
O therwise.
Obtain Var(30*Y1+ 25*Y2).

Thank you very much.

Originally Posted by Ruichan

1. Let Y1 and Y2 be independent random variables that are both uniformly distributed on the interval (0,1). Find P(Y1< 2*Y2|Y1< 3*Y2).

The event [(Y1<2*Y2) or (Y1<3*Y2)] is the same as the event (Y1<3*Y3).

This is because the event (Y1<2*Y2) coresponds to points above the line
Y2=(1/2)Y1 in the Y1, Y2 plane, and inside the unit square, and the event
(Y1<3*Y2) coresponds to points above the line Y2=(1/3)Y1 in the Y1, Y2
plane, and inside the unit square. The latter region includes all of the first
region.

Also as Y1 and Y2 are uniformly distributed on the unit square the required
probability corresponds to the area of the the region, which is:

1-(1/6)=5/6.

(the area not in the region is a triangle of base 1 and altitude 1/3).

RonL