Probability 3.1 Discrete Random Variables Basics 2 Anna Karlin Most slides by Alex Tsun
Agenda ● Recap on rvs ● Expectation ● Linearity of Expectation (LoE) ● Law of the Unconscious Statistician (Lotus)
Random Variable
Probability Mass Function (PMF)
Homeworks of 3 students returned randomly were Hw ● Each permutation equally likely ● X: # people who get their own homework r Ax 231 Prob Outcome w X(w) 312 0 7 132 1/6 1 2 3 3 I 213 1/6 1 3 2 1 73 321 123 1/6 2 1 3 1 1/6 2 3 1 0 1/6 3 1 2 0 PMI CD 1/6 3 2 1 1 t F cxkPr Xex qPxM
Expectation
Homeworks of 3 students returned randomly ● Each permutation equally likely ● X: # people who get their own homework ● What is E(X)? r NX PMI 23 Prob Outcome w X(w) go E k o 1/6 1 2 3 3 p lH ta 1 213 73 1/6 1 3 2 1 321 wise 123 1/6 2 1 3 1 1/6 2 3 1 0 to'zt3 to O zt F X p Ck 1/6 3 1 2 0 1/6 3 2 1 1 Hw Pfw wer
Repeated coin flipping Flip a biased coin with probability p of coming up Heads n times. Each flip independent of all others. X is number of Heads. What is E(X)? EIxt E.d.FI p Ckf P X k7 pi i.n dy n 5 E pIPogqyu pram p5 1417410445 HTT HTT
Repeated coin flipping Flip a biased coin with probability p of coming up Heads n times. X is number of Heads. What is E(X)?
Linearity of Expectation (Idea) Let’s say you and your friend sell fish for a living. ● Every day you catch X fish, with E[X] = 3 . ● Every day your friend catches Y fish, with E[Y] = 7 . how many fish do the two of you bring in ( Z = X + Y ) on an average day? E[Z] = E[X + Y] = e[X] + E[Y] = 3 + 7 = 10 1 ELY 10 7 F X 3 You can sell each fish for $5 at a store, but you need to pay $20 in rent. How much profit do you expect to make? E[5Z - 20] = 5E[Z] - 20 = 5 x 10 - 20 = 30
Linearity of Expectation (Idea) Let’s say you and your friend sell fish for a living. ● Every day you catch X fish, with E[X] = 3 . ● Every day your friend catches Y fish, with E[Y] = 7 . how many fish do the two of you bring in ( Z = X + Y ) on an average day? E[Z] = E[X + Y] = You can sell each fish for $5 at a store, but you need to pay $20 in rent. How much profit do you expect to make? E[5Z - 20] = 5E[Z] - 20 = 5 x 10 - 20 = 30
Linearity of Expectation (Idea) Let’s say you and your friend sell fish for a living. ● Every day you catch X fish, with E[X] = 3 . ● Every day your friend catches Y fish, with E[Y] = 7 . how many fish do the two of you bring in ( Z = X + Y ) on an average day? E[Z] = E[X + Y] = e[X] + E[Y] = 3 + 7 = 10 You can sell each fish for $5 at a store, but you need to pay $20 in rent. How much profit do you expect to make? E[5Z - 20] = 5E[Z] - 20 = 5 x 10 - 20 = 30 SECY 20 20 30 5 lo
Linearity of Expectation (Idea) Let’s say you and your friend sell fish for a living. ● Every day you catch X fish, with E[X] = 3 . ● Every day your friend catches Y fish, with E[Y] = 7 . how many fish do the two of you bring in ( Z = X + Y ) on an average day? E[Z] = E[X + Y] = e[X] + E[Y] = 3 + 7 = 10 You can sell each fish for $5 at a store, but you need to pay $20 in rent. How much profit do you expect to make? E[5Z - 20] = 5E[Z] - 20 = 5 x 10 - 20 = 30
Linearity of Expectation (LoE) i
Linearity of Expectation (Proof)
<latexit sha1_base64="CxIQlQmNr6yr2qKRf1V6LcRsARQ=">ACKHicbVBdS8MwFE39nPOr6qMvwSGsCKOdgr6IQxF8nOC2wlZKmVbWJqWJBVG2c/xb/i4gie/WXmHZ7mJsXEs4951ySe4KYUalse2KsrK6tb2wWtorbO7t7+bBYVNGicCkgSMWCTdAkjDKSUNRxYgbC4LCgJFWMLzL9NYzEZJG/EmNYuKFqM9pj2KkNOWbN/dl13fgGXT9qr47rBspmbfcgte5aOk2A1VozTsyilu+WbIrdl5wGTgzUAKzqvmR6cb4SQkXGpGw7dqy8FAlFMSPjYieRJEZ4iPqkrSFHIZFemi86hqea6cJeJPThCubs/ESKQilHYaCdIVIDuahl5H9aO1G9Ky+lPE4U4Xj6UC9hUEUwSw12qSBYsZEGCAuq/wrxAmElc62qENwFldeBs1qxTmvVB8vSrXbWRwFcAxOQBk4BLUwAOogwbA4AW8gU/wZbwa78a3MZlaV4zZzBH4U8bPL02Kn2w=</latexit> Corollary: linearity for sum of lots of r.v.s E ( X 1 + X 2 + . . . + X n ) = E ( X 1 ) + E ( X 2 ) + . . . + E ( X n ) Proof by induction!
Homeworks of students returned randomly ● Each permutation equally likely ● X: # people who get their own homework EEkPCx EH ● What is E(X) when there are n students? Prob Outcome w X X X 2 3 is EtE e II 1/6 1 2 3 3 Xi 1/6 1 3 2 1 I 0 O otherwise o O 1/6 2 1 3 1 I O 1/6 2 3 1 0 O O 0 1/6 3 1 2 0 O O O tX X 0 1/6 3 2 1 1 O I F xt ECX.tl atXs O prfxi otzprlx.at F Xi tE XDtEH3 E X I Pr Xi 1
Indicator random variable ● For any event A, can define the indicator random variable for A F XA O Prf E H PRCA prfA Ai event that Xi studenti HW back got own
Computing complicated expectations ● Often boils down to finding the right way to decompose the random variable into simple random variables (often indicator random variables) and then applying linearity of expectation.
Repeated coin flipping Flip a biased coin with probability p of coming up Heads n times. X is number of Heads. What is E(X)?
Repeated coin flipping times n independently Flip a biased coin with probability p of coming up Heads n times. X is number of Heads. tXn X that X T Ind Sd What is E(X)? ng n Y I Xi Pds Pr inflipis A Eui p otherwise 0 _E X that.tl tECXn F X ECXDtECxa n t p p p np
I m Pairs with same birthday d 365 euhstudent uniform sample space has random indep bday ● In a class of m students, on average how many pairs of people have the same birthday? people with bday X pairs of same E k Prix _k F X IT k o names of people m 1,2 ki are snm s EKE Eas I Xii same bday o w 0 345 127345 IEDxt EEE.gl ii p E j Prf have sambday students students GS E prfshdntnistafdubdaydahbdayondayl 3 65 ITS 6503165
F G I B E B F Rotating the table G C D A ra n people are sitting around a circular table. There is a nametag in each place Nobody is sitting in front of their own nametag. t Rotate the table by a random number k of positions between 1 and n-1 (equally likely). X is the number of people that end up front of their own nametag. What is E(X)? X X tXat tX YingsgyEginatum's x4Xi EHHEH.lt E Xn t O t ow IF ECXif prfpusmiwp.ee'Entytheanae
r Ux 231 312 0 132 I 213 7 3 321 123
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