9 in stata di binomialtail1222 72512209 di 1

Info iconThis preview shows pages 9–18. Sign up to view the full content.

View Full Document Right Arrow Icon
9
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
In Stata: . di binomialtail(12,2,.2) .72512209 . di 1 - binomial(12,1,.2) .72512209 The command di is short for “display.” So there is a very good chance a coalition of two will form convinced of the defendent’s innocence. 10
Background image of page 10
Later we will provide simple proofs that E X np Var X np 1 p which reduce to the Bernoulli case when n 1. By being a bit clever with summations, and using the fact that a discrete density must sum to unity over all values, E X can be derived as follows: 11
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
E X k 0 n k n k p k 1 p n k k 1 n k n ! k ! n k ! p k 1 p n k k 1 n n ! k 1 ! n k ! p k 1 p n k np k 1 n n 1 ! k 1 ! n k ! p k 1 1 p n k Do a change of variables: h k 1, so h ranges from 0 to k 1. Inserting k 1 h and k h 1 we have 12
Background image of page 12
k 1 n n 1 ! k 1 ! n k ! p k 1 1 p n k h 0 n 1 n 1 ! h ! n 1 h ! p h 1 p n 1 h 1. The last equality follows because each summand in the final expression is the density for a Binomial n 1, p distribution, and therefore the PDF sums to unity over all possible values. It now follows that E X np . A similar but more tedious argument works for E X 2 np 1 p np 2 . 13
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
EXAMPLE : Suppose an airline is trying to decide how many reservations to take for a flight that has 100 seats. Any particular passenger keeps his or her reservation with probability p .80. Ignore the fact that some passengers travel in group and assume that whether someone shows up for the flight is independent of everyone else’s decision. If n reservations are made and X is the number of passengers showing up then X ~ Binomial n ,.80 . 14
Background image of page 14
If the airline makes n 120 reservations, it expects E X 120 .80 96 passengers to show up. But the probability that more than 100 passengers show up is . di 1 - binomial(120,100,.8) .15171351 . di binomialtail(120,101,.8) .15171351 15
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
What if the airline wants to maximize expected profits? Suppose each passenger generates r dollars in revenue, but it costs the airline c dollars for each overbooked passenger. (That is, the airline must compensate passengers who are unable to get on the flight.) We can write the profit function as g x rx if x 100 100 r x 100 c if x 100 If we let Y denote (random) profits, then Y g X . 16
Background image of page 16
For any number of reservations n , we can write expected profit as E g X  x 0 n g x n x  .8 x .2 n x x 0 100 rx n x x n x x 101 n 100 r x 100 c n x x n x Given r and c we can, in principle, find the value of n that maximizes expected profits.
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 18
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page9 / 89

9 In Stata di binomialtail1222 72512209 di 1 binomial1212...

This preview shows document pages 9 - 18. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online