HW7 P3 supplement

# HW7 P3 supplement - realization or observed value of RV S...

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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
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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 ( = > P . Now note that for X > t, S = 0. Since , t / 1 s > ≥ 1 ) t X | s S ( = > ≤ 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 − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ....
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## This note was uploaded on 01/16/2010 for the course ECE 3100 at Cornell.

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