Stat 31, Section 1, Last Time Big Rules of Probability –The not rule –The or rule –The and rule P{A & B} = P{A|B}P{B} = P{B|A}P{A} Bayes Rule (turn around Conditional Probabilities) Independence

(Need one more major concept at this level) An event A does not depend on B, when: Knowledge of B does not change chances of A: P{A | B} = P{A}

New Ball & Urn Example H  R R R R G G T  R R G Again toss coin, and draw ball: Same, so R & I are independent events Not true above, but works here, since proportions of R & G are same

Independence Note, when A in independent of B: so And thus i.e. B is independent of A

Independence Note, when A in independent of B: It follows that: B is independent of A I.e. “independence” is symmetric in A and B (as expected) More formal treatments use symmetric version as definition (to avoid hassles with 0 probabilities)

Independence HW: 4.33

Special Case of “And” Rule For A and B independent: P{A & B} = P{A | B} P{B} = P{B | A} P{A} = = P{A} P{B} i.e. When independent, just multiply probabilities…

Independent “And” Rule E.g. Toss a coin until the 1 st Head appears, find P{3 tosses}: Model: tosses are independent (saw this was reasonable last time, using “equally likely sample space ideas) P{3 tosses} = When have 3: group with parentheses

Independent “And” Rule E.g. Toss a coin until the 1 st Head appears, find P{3 tosses} (by indep:) I.e. “just multiply”

Independent “And” Rule E.g. Toss a coin until the 1 st Head appears, P{3 tosses} Multiplication idea holds in general So from now on will just say: “Since Independent, multiply probabilities” Similarly for Exclusive Or rule, Will just “add probabilities”

Independent “And” Rule HW:

Overview of Special Cases Careful: these can be tricky to keep separate OR works like adding, for mutually exclusive AND works like multiplying, for independent

Overview of Special Cases Caution: special cases are different Mutually exclusive independent For A and B mutually exclusive: P{A | B} = 0 P{A} Thus not independent

Overview of Special Cases HW: C13 Suppose events A, B, C all have probability 0.4, A & B are independent, and A & C are mutually exclusive. (a)Find P{A or B} (0.64) (b)Find P{A or C} (0.8) (c)Find P{A and B} (0.16) (d)Find P{A and C} (0)

Random Variables Text, Section 4.3 (we are currently jumping) Idea: take probability to next level Needed for probability structure of political polls, etc.

Random Variables Definition: A random variable, usually denoted as X, is a quantity that “takes on values at random”

Random Variables Two main types (that require different mathematical models) Discrete, i.e. counting (so look only at “counting numbers”, 1,2,3,…) Continuous, i.e. measuring (harder math, since need all fractions, etc.)

Random Variables E.g: X = # for Candidate A in a randomly selected political poll: discrete (recall all that means) Power of the random variable idea: Gives something to “get a hold of…” Similar in spirit to high school algebra: Give unknowns a name, so can work with

Random Variables E.g: X = # that comes up, in die rolling: Discrete But not too interesting Since can study by simple methods As done above Don’t really need random variable concept

Random Variables E.g: X = # that comes up, in die rolling: Discrete But not very interesting Since can study by simple methods As done above Don’t really need random variable concept

Random Variables E.g: Measurement error: Let X = measurement: Continuous How to model probabilities???

Random Variables HW on discrete vs. continuous: 4.40 ((b) discrete, (c) continuous, (d) could be either, but discrete is more common)

And now for something completely different My idea about “visualization” last time: 30% really liked it 70% less enthusiastic… Depends on mode of thinking –“Visual thinkers” loved it –But didn’t connect with others So don’t plan to continue that…

Random Variables A die rolling example (where random variable concept is useful) Win $9 if 5 or 6, Pay $4, if 1, 2 or 3, otherwise (4) break even Notes: Don’t care about number that comes up Random Variable abstraction allows focussing on important points Are you keen to play? (will calculate…)

Random Variables Die rolling example Win $9 if 5 or 6, Pay $4, if 1, 2 or 4 Let X = “net winnings” Note: X takes on values 9, -4 and 0 Probability Structure of X is summarized by: P{X = 9} = 1/3 P{X = -4} = ½ P{X = 0} = 1/6 (should you want to play?, study later)

Random Variables Die rolling example, for X = “net winnings”: Win $9 if 5 or 6, Pay $4, if 1, 2 or 4 Probability Structure of X is summarized by: P{X = 9} = 1/3 P{X = -4} = ½ P{X = 0} = 1/6 Convenient form: a table Winning9-40 Prob.1/31/21/6

Summary of Prob. Structure In general: for discrete X, summarize “distribution” (i.e. full prob. Structure) by a table: Where: i.All are between 0 and 1 ii. (so get a prob. funct’n as above) Valuesx1x1 x2x2 …xkxk Prob.p1p1 p2p2 …pkpk

Summary of Prob. Structure Summarize distribution, for discrete X, by a table: Power of this idea: Get probs by summing table values Special case of disjoint OR rule Valuesx1x1 x2x2 …xkxk Prob.p1p1 p2p2 …pkpk

Summary of Prob. Structure E.g. Die Rolling game above: P{X = 9} = 1/3 P{X < 2} = P{X = 0} + P{X = -4} = 1/6 +½ = 2/3 P{X = 5} = 0 (not in table!) Winning9-40 Prob.1/31/21/6

Summary of Prob. Structure E.g. Die Rolling game above: Winning9-40 Prob.1/31/21/6

Summary of Prob. Structure HW: 4.47 & (d) Find P{X = 3 | X >= 2} (0.24) 4.50 (0.144, …, 0.352)

Random Variables Now consider continuous random variables Recall: for measurements (not counting) Model for continuous random variables: Calculate probabilities as areas, under “probability density curve”, f(x)

Continuous Random Variables Model probabilities for continuous random variables, as areas under “probability density curve”, f(x): = Area( ) a b (calculus notation)

Continuous Random Variables e.g. Uniform Distribution Idea: choose random number from [0,1] Use constant density: f(x) = C Models “equally likely” To choose C, want: Area 1 = P{X in [0,1]} = C So want C =

Uniform Random Variable HW: 4.52 (0.73, 0, 0.73, 0.2, 0.5) 4.54 (1, ½, 1/8)

Continuous Random Variables e.g. Normal Distribution Idea: Draw at random from a normal population f(x) is the normal curve (studied above) Review some earlier concepts:

Normal Curve Mathematics The “normal density curve” is: usual “function” of circle constant = 3.14… natural number = 2.7…

Normal Curve Mathematics Main Ideas: Basic shape is: “Shifted to mu”: “Scaled by sigma”: Make Total Area = 1: divide by as, but never

Computation of Normal Areas EXCEL Computation: works in terms of “lower areas” E.g. for Area < 1.3

Computation of Normal Probs EXCEL Computation: probs given by “lower areas” E.g. for X ~ N(1,0.5) P{X < 1.3} = 0.73

Normal Random Variables As above, compute probabilities as areas, In EXCEL, use NORMDIST & NORMINV E.g. above: X ~ N(1,0.5) P{X < 1.3} =NORMDIST(1.3,1,0.5,TRUE) = 0.73 (as in pic above)

Normal Random Variables HW: 4.55