Lecture 2: Probability, Random Variables and Probability Distributions. GENOME 560, Spring 2015 Doug Fowler, GS

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1 Lecture 2: Probability, Radom Variables ad Probability Distributios GENOME 560, Sprig 2015 Doug Fowler, GS 1

2 Course Aoucemets Problem Set 1 will be posted Due ext Thursday before class Please go to the course website spr16/home?pli=1 Access the Catalyst dropbox from there. 2

3 Brief Review of Last Lecture Types of data Descriptive statistics 3

4 Outlie Itroductio to probability Radom variables Discrete ad cotiuous probability distributios R exercises How to use R for calculatig descriptive statistics ad makig graphs 4

5 What is probability? 5

6 What is probability? Likeliess that a evet will occur More formally, the extet to which a evet is likely to occur, measured by the ratio of the favorable cases to the whole umber of cases possible 6

7 Sample Spaces A sample space Ω is the set of all possible outcomes of a experimet sample space Ω 7

8 Sample Spaces A sample space Ω is the set of all possible outcomes of a experimet sample space A particular outcome Ω 8

9 Sample Spaces - Examples Whe flippig a coi oce A sigle roll of a d6 A sigle roll of two dice 9

10 Sample Spaces - Examples Whe flippig a coi oce A sigle roll of a d6 A sigle roll of two dice 10

11 Sample Spaces - Examples Whe flippig a coi oce A sigle roll of a d6 A sigle roll of two dice 11

12 Sample Spaces - Examples Whe flippig a coi oce A sigle roll of a d6 A sigle roll of two dice 12

13 Evets A evet E is a outcome or set of outcomes ad is a subset of Ω with a defied umerical probability Gettig heads o a sigle toss Gettig a eve-valued die roll Gettig oe heads i two coi tosses 13

14 Evets A evet E is a outcome or set of outcomes ad is a subset of Ω with a defied umerical probability Gettig heads o a sigle toss Gettig a eve-valued die roll Gettig oe heads i two coi tosses 14

15 Rules of Probability The probability of a evet A occurrig is deoted Pr(A) ad is a measure of certaity that A will occur, subject to: 15

16 Rules of Probability The probability of a evet A is deoted Pr(A) ad is a measure of certaity that A will occur, subject to: Somethig i Ω must occur 16

17 Rules of Probability The probability of a evet A is deoted Pr(A) ad is a measure of certaity that A will occur, subject to: Somethig i Ω must occur A c is the complemet of A 17

18 Rules of Probability The probability of a evet A is deoted Pr(A) ad is a measure of certaity that A will occur, subject to: Somethig i Ω must occur A c is the complemet of A Additio rule for disjoit (mutually exclusive) evets 18

19 Rules of Probability The probability of a evet A is deoted Pr(A) ad is a measure of certaity that A will occur, subject to: Somethig i Ω must occur A c is the complemet of A Additio rule for disjoit (mutually exclusive) evets Multiplicatio rule for idepedet evets 19

20 Joit ad Margial Probabilities Let s say we have two loci (L1 ad L2), each with two alleles (A1, B1 ad A2, B2) We re iterested i uderstadig the probability of the alleles occurrig joitly at the two loci (e.g. their joit probability) 20

21 Joit ad Margial Probabilities Let s say we have two loci (L1 ad L2), each with two alleles (A1, B1 ad A2, B2) P(L1 = A1) P(L1 = B1) P(L2) P(L2 = A2) P(L2 = B2) P(L1) We ca arrage the probability of fidig each allele at each locus i a table 21

22 Joit ad Margial Probabilities The margis of the table give the probabilities for each locus without cosiderig the other oe P(L1 = A1) P(L1 = B1) P(L2) P(L2 = A2) P(L2 = B2) P(L1) This is what you would fid if you sampled just at L1 or L2 aloe 22

23 Joit ad Margial Probabilities The middle of the table gives the joit probability distributio P(L1 = A1) P(L1 = B1) P(L2) P(L2 = A2) P(L2 = B2) P(L1) This is what you would fid if samplig at both loci simultaeously 23

24 Coditioal Probabilities Coditioal probability expresses the depedece of oe evet o aother P(L1 = A1) P(L1 = B1) P(L2) P(L2 = A2) P(L2 = B2) P(L1)

25 Coditioal Probabilities Coditioal probability expresses the depedece of oe evet o aother P(L1 = A1) P(L1 = B1) P(L2) P(L2 = A2) P(L2 = B2) P(L1) Notice that if we fid A1 at locus 1 we are more likely to fid B2 at locus 2 tha if we fid B1 at locus 1. So, these evets are depedet 25

26 Coditioal Probability Give the joit probabilities, we ca write dow the coditioal probability for ay of the evets A AB B 26

27 Coditioal Probability The coditioal probability of a evet A give that B has occurred: If A ad B are idepedet evets what is the? 27

28 Bayes Theorem From here, we ca derive Bayes theorem posterior probability prior probability Bayes theorem describes the probability of a evet (A) occurrig the cotext of aother, possibly related evet (B) 28

29 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: 29

30 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: Let s say I kow this particular fluorescet fusio protei is toxic. I m iterested to kow what the probability is that a dead cell is bright. After all, this could really screw up my experimet 30

31 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: What s a reasoable assumptio I could make about the fractio of dead cells that will be bright? 31

32 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: What s a reasoable assumptio I could make about the fractio of dead cells that will be bright? Well, it would be reasoable to assume that 70% of dead cells will be bright. But, it could be that dead cells do ot have the same distributio of bright/dim/nf as live oes. 32

33 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: I flow sort with a vital dye to lear probability of a cell from each category beig dead: 33

34 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: I flow sort with a vital dye to lear probability of a cell from each category beig dead: I fact, I lear that dead cells have a very differet distributio of bright/dim/nf tha live oes 34

35 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: I flow sort with a vital dye to lear probability of a cell from each category beig dead: Now, I ca use Bayes rule to update my kowledge about the fractio of dead cells that are bright. 35

36 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: I flow sort with a vital dye to lear probability of a cell from each category beig dead: What is the probability of a dead cell beig bright? 36

37 Bayes Theorem - Example I am lookig at yeast cells expressig a fluorescet protei ad I classify them ito three categories: bright, dim ad ofluorescet. I observe that: prior probability I flow sort with a vital dye to lear probability of a cell from each category beig dead: experimet What is the probability of a dead cell beig bright? posterior probability 37

38 Sample Spaces A sample space Ω is the set of all possible outcomes of a experimet sample space A particular outcome Ω 38

39 Evets i sample space map to values However, evets themselves are t useful if we wat to do math (e.g. what s the meaig of heads or a cube with five dots o it ) sample space Ω We ca relate evets to umerical outcomes 39

40 Radom Variables (RV) A RV is a variable whose value results from the measuremet of a quatity that is subject to variatios due to chace (i.e. radomess). e.g. dice throwig outcome, expressio level of gee A More formally 40

41 Radom Variables (RV) A variable whose value is a umerical outcome of a experimet RVs is a fuctio that maps from evets to umerical values (e.g. heads = 1, tails = 0) NOT AN VARIABLE AS IN ALGEBRA (is a fuctio) 41

{1,2,3,4,5,6} A RV is the fuctio that associates each outcome with a umber evets Let s cosider a

42 What Does That Mea? Say that you throw a die There are 6 possible outcomes (or evets) Associate each evet with a umber {1,2,3,4,5,6} A RV is the fuctio that associates each outcome with a umber evets Let s cosider a expressio level of gee A There are may possible evets (actual trascript umber) RV associates each evet with a cotiuous-valued umber represetig expressio level of gee A (i, say RPKM) 42

43 Two Types of Radom Variables A discrete RV has a coutable umber of possible values e.g. dice throwig outcome, geotype of a SNP, etc A cotiuous RV ca take o all values i a iterval of umbers e.g. expressio level gee A, blood glucose level, etc 43

44 Probability Distributio of Discrete RVs Discrete Let X be a discrete RV. The the probability mass fuctio (pmf), f(x), of X is: 44

45 Probability Distributio of Discrete RVs Discrete Let X be a discrete RV. The the probability mass fuctio (pmf), f(x), of X is: The pmf returs the probability of the RV X takig o a value of x, if x is a elemet of the sample space 45

46 Probability Distributio of Discrete RVs Discrete Let X be a discrete RV. The the probability mass fuctio (pmf), f(x), of X is: The pmf returs the probability of the RV X takig o a value of x, if x is a elemet of the sample space If x is ot i the sample space, the pmf is 0 46

47 Probability Distributio of Discrete RVs Discrete Let X be a discrete RV. The the probability mass fuctio (pmf), f(x), of X is: Example let X be the result of a fair coi flip, X=0 for tails, X=1 for heads 47

48 Probability Distributio of Discrete RVs Discrete Let X be a discrete rv. The the probability mass fuctio (pmf), f(x), of X is: Example let X be the result of a coi flip, X=0 for tails, X=1 for heads H T X = Coi toss outcome 48

49 Probability Distributio of Discrete RVs Discrete Let X be a discrete RV. The the probability mass fuctio (pmf), f(x), of X is: Example let X be the result of a coi flip, X=0 for tails, X=1 for heads ow suppose we do t kow if the coi is fair we ca itroduce a parameter θ to represet the Pr(head) H T X = Coi toss outcome 49

50 Distributios defied by parameters are importat! H T X = Coi toss outcome If we ca assume that X has a particular distributio, ad we kow the parameters the we ca calculate whatever we wat (mea/variace) If we ca write dow a parametric distributio for X we ca lear the parameters from the data (max likelihood, etc) Parametric iferetial statistics (e.g. learig about populatios) is all about comparig parameters 50

51 Probability Dist of Cotiuous RVs Cotiuous Let X be a cotiuous RV. The the probability desity fuctio (pdf) of X is a fuctio f(x) such that for ay two umbers a ad b with a b 51

52 Probability Dist of Cotiuous RVs Cotiuous Let X be a cotiuous RV. The the probability desity fuctio (pdf) of X is a fuctio f(x) such that for ay two umbers a ad b with a b Example The time i years from diagosis util death of a patiet with a specific cacer has the PDF: desity survival time (years) 52

53 Probability Dist of Cotiuous RVs Cotiuous Let X be a cotiuous RV. The the probability desity fuctio (pdf) of X is a fuctio f(x) such that for ay two umbers a ad b with a b Example The time i years from diagosis util death of a patiet with a specific cacer has the PDF: What is the chace of death i years 3-5? desity survival time (years) 53

54 Probability Dist of Cotiuous RVs Cotiuous Let X be a cotiuous RV. The the probability desity fuctio (pdf) of X is a fuctio f(x) such that for ay two umbers a ad b with a b Example The time i years from diagosis util death of a patiet with a specific cacer has the PDF: What is the chace of death i years 3-5? Itegrate curve from 3->5 = 0.18 desity survival time (years) 54

55 Expectatio of Radom Variables Ituitively, the expected value of a RV is the log-ru average of repetitios of the experimet Previous coi-flip example (X=1, heads; X=0, tails) 1/*( ) = 0.5 The expectatio value is also equal to the populatio mea μ 55

56 Expectatio of Radom Variables Discrete Let X be a discrete RV that takes o values i the set D ad has a pmf f(x). The the expected or mea value of X is: 56

57 Expectatio of Radom Variables Discrete Let X be a discrete RV that takes o values i the set D ad has a pmf f(x). The the expected or mea value of X is: For example, let s say that X is a RV represetig the outcome of a die throw X ca be 1, 2, 3, 4, 5, or 6; so D = {1,2,3,4,5,6} What is the expected value of X? 57

The the expected or mea value of X is: For example, let s say that X is a RV

58 Expectatio of Radom Variables Discrete Let X be a discrete RV that takes o values i the set D ad has a pmf f(x) that gives Pr(X=x). The the expected or mea value of X is: For example, let s say that X is a RV represetig the outcome of a die throw X ca be 1, 2, 3, 4, 5, or 6; so D = {1,2,3,4,5,6} What is the expected value of X? 58

59 Expectatio of Radom Variables Cotiuous The expected or mea value of a cotiuous RV X with pdf f(x) is: 59

60 Law of Large Numbers As the umber of observatios i a sample icreases, the sample mea approaches the expected value/populatio mea This is ot, of course, because the pmf/pdf chages (e.g. tails does ot become more likely because we get a log ru of heads) 60

61 Variace of Radom Variables Discrete Let X be a discrete RV with pmf f(x) ad expected value μ. The variace of X is: Cotiuous The variace of a cotiuous rv X with pdf f(x) ad mea μ is: 61

62 Example of Expectatio ad Variace Let L 1, L 2,, L be a sequece of ucleotides ad defie the RV X i as: 62

63 Example of Expectatio ad Variace Let L 1, L 2,, L be a sequece of ucleotides ad defie the RV X i as: pmf is the: 63

64 Example of Expectatio ad Variace Let L 1, L 2,, L be a sequece of ucleotides ad defie the RV X i as: pmf is the: 64

65 Example of Expectatio ad Variace Let L 1, L 2,, L be a sequece of ucleotides ad defie the RV X i as: pmf is the: 65

66 Example of Expectatio ad Variace Let L 1, L 2,, L be a sequece of ucleotides ad defie the RV X i as: pmf is the: 66

67 A big(ish) data set of your very ow

68 Deep mutatioal scaig to measure protei fuctio variat score Erich2

69 A big(ish) data set of your very ow Colum ame Descriptio positio_id Uique idetifier of the positio i the protei variat_id Uique idetifier of a variat dms_id Uique idetifier of the DMS orgaism Orgaism of origi for the protei uiprot_id ID i the Uiprot databade reported_effect Log base 2 fuctioal score reported by the authors scaled_effect reported_effect, scaled aa1 Idetity of the WT amio acid at the variat positio aa2 idetify of the mutat amio acid positio Positio at which the mutatio occurred aa1_polarity Polarity of WT amio acid aa2_polarity Polarity of mutat amio acid aa1_pi Isoelectric poit of WT amio acid aa2_pi Isoelectric poit of mutat amio acid delta_pi aa1_pi - aa2_pi aa1_weight Molecular weight of WT amio acid aa2_weight Molecular weight of mutat amio acid delta_weight aa1_weight - aa2_weight aa1_volume Volume of WT amio acid aa2_volume Volume of mutat amio acid delta_volume aa1_volume - aa2_volume aa1_psic PSIC score of WT amio acid (based o multiple sequece aligmet high = less damagig, low = more damagig) PSIC score of mutat amio acid (based o multiple sequece aligmet high = less damagig, low = more aa2_psic damagig) delta_psic aa1_psic - aa2_psic

70 R exercises How to use R for calculatig descriptive statistics ad makig graphs We will use published microarray expressio data You will be usig these data for your Problem Set 1! Dowload from MA_Filtered.txt Read it from R by doig: ww_data = read.table(file=" OM560/560_ww_data.txt", header = T, sep = '\t') 70

Lecture 2: Probability, Random Variables and Probability Distributions. GENOME 560, Spring 2017 Doug Fowler, GS

Lecture 2: Probability, Random Variables and Probability Distributions. GENOME 560, Spring 2017 Doug Fowler, GS Lecture 2: Probability, Radom Variables ad Probability Distributios GENOME 560, Sprig 2017 Doug Fowler, GS (dfowler@uw.edu) 1 Course Aoucemets Problem Set 1 will be posted Due ext Thursday before class