HW7 P3 supplement - realization, or observed value, of RV...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Several people were confused about problem 3b on HW 7, so I’ll attempt to write up a more detailed solution. We want to calculate for all s. We will accomplish this by calculating for the following three regions: ) s S ( ) s ( F S = P ) s ( F S 1) 0 s < 2) t / 1 s 0 < 3) t / 1 s Region 1: Here, . Since , and X is non-negative, the random variable S can only take on non- negative values. Thus, in this region. 0 s < > = t X 0 t X X / 1 S 0 ) s S ( ) s ( F S = = P Region 2: Here, . So t / 1 s 0 < ) s / 1 X ( ) s X / 1 ( ) s S ( ) s ( F S = = = P P P . However, since , so the event t / 1 s 0 < < < s / 1 t {} { = > = t X s / 1 X } {Alvin’s hit is outside the inner circle}. Thus, 2 S r t 1 ) t X ( 1 ) t X ( ) s ( F = = > = P P . Region 3: Here, s . By total probability, we have t / 1 ) t X | s S ( r t 1 ) t X | s S ( r t ) t X ( ) t X | s S ( ) t X ( ) t X | s S ( ) s S ( ) s ( F 2 2 S > + = > > + = = P P P P P P P . It is important to note here that over this region (and the other two regions), we are limiting s, which is a
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: realization, or observed value, of RV S, and that we are not limiting the RV S . Several people mistakenly thought that we were letting S in this region, and from that, concluded that , and thus, t / 1 t X ) t X ( = &gt; P . Now note that for X &gt; t, S = 0. Since , t / 1 s &gt; 1 ) t X | s S ( = &gt; P . 2 2 t s 1 1 ) t X ( ) t X s / 1 ( ) t X ( ) t X , s / 1 X ( ) t X | s X / 1 ( ) t X | s S ( = = = = P P P P P P . Thus, 2 2 2 2 2 2 S r s 1 1 r t 1 t s 1 1 r t ) s ( F = + = ....
View Full Document

Ask a homework question - tutors are online