Basic Ideas in Probability and Statistics for Experimenters
Basic Ideas in Probability and Statistics for Experimenters: Part I: Qualitative Discussion He uses statistics as a drunken man uses lamp-posts – for support rather than for illumination … A. Lang Shivkumar Kalyanaraman Rensselaer Polytechnic Institute shivkuma@ecse. rpi. edu http: //www. ecse. rpi. edu/Homepages/shivkuma Based in part upon. Shivkumar slides of Prof. Kalyanaraman Raj Jain (OSU) Rensselaer Polytechnic Institute 1
Overview Why Probability and Statistics: The Empirical Design Method… q Qualitative understanding of essential probability and statistics q Especially the notion of inference and statistical significance q Key distributions & why we care about them… q Reference: Chap 12, 13 (Jain), Chap 2 -3 (Box, Hunter), and q q http: //mathworld. wolfram. com/topics/Probabilityand. Statistics. html Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 2
Why Care About Prob. & Statistics? How to make this empirical design process EFFICIENT? ? How to avoid pitfalls in inference! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 3
Probability Think of probability as modeling an experiment q The set of all possible outcomes is the sample space: S q q Classic “Experiment”: Tossing a die: S = {1, 2, 3, 4, 5, 6} q Any subset A of S is an event: A = {the outcome is even} = {2, 4, 6} Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 4
Probability of Events: Axioms • P is the Probability Mass function if it maps each event A, into a real number P(A), and: i. ) ii. ) P(S) = 1 iii. )If A and B are mutually exclusive events then, Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 5
Probability of Events …In fact for any sequence of pair-wise-mutually -exclusive events, we have Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 6
Other Properties q q Derived by breaking up above sets into mutually exclusive pieces and comparing to fundamental axioms!! Rensselaer Polytechnic Institute 7 Shivkumar Kalyanaraman
Conditional Probability = (conditional) probability that the outcome is in A given that we know the outcome in B • • Example: Toss one die. • Note that: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 8
Independence q q q Events A and B are independent if P(AB) = P(A)P(B). Also: and Example: A card is selected at random from an ordinary deck of cards. q A=event that the card is an ace. q B=event that the card is a diamond. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 9
Random Variable as a Measurement q We cannot give an exact description of a sample space in these cases, but we can still describe specific measurements on them q. The temperature change produced. q. The number of photons emitted in one millisecond. q. The time of arrival of the packet. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 10
Random Variable as a Measurement q Thus a random variable can be thought of as a measurement on an experiment Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 11
Probability Mass Function for a Random Variable q The probability mass function (PMF) for a (discrete valued) random variable X is: q Note that q Also for a (discrete valued) random variable X for Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 12
Probability Distribution Function (pdf) a. k. a. frequency histogram, p. m. f (for discrete r. v. ) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 13
Cumulative Distribution Function q The cumulative distribution function (CDF) for a random variable X is q Note that q Also is non-decreasing in x, i. e. and Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 14
PMF and CDF: Example Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 15
Expectation of a Random Variable q The expectation (average) of a (discrete-valued) random variable X is q Three coins example: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 16
Expectation (Mean) = Center of Gravity Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 17
Median, Mode q Median = F-1 (0. 5), where F = CDF q Aka 50% percentile element q I. e. Order the values and pick the middle element q Used when distribution is skewed q Mode: Most frequent or highest probability value q Multiple modes are possible q Need not be the “central” element q Mode may not exist (eg: uniform distribution) q Used with categorical variables Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 18
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Measures of Spread/Dispersion: Why Care? You can drown in a river of average depth 6 inches! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 21
Standard Deviation, Coeff. Of Variation, SIQR Variance: second moment around the mean: q 2 = E((X- )2) q Standard deviation = q Coefficient of Variation (C. o. V. )= / q SIQR= Semi-Inter-Quartile Range (used with median = 50 th percentile) q (75 th percentile – 25 th percentile)/2 q Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 22
Covariance and Correlation: Measures of Dependence q Covariance: q q q = For i = j, covariance = variance! Independence => covariance = 0 (not vice-versa!) Correlation (coefficient) is a normalized (or scaleless) form of covariance: q Between – 1 and +1. q Zero => no correlation (uncorrelated). q Note: uncorrelated DOES NOT mean independent! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 23
Continuous-valued Random Variables So far we have focused on discrete(-valued) random variables, e. g. X(s) must be an integer q Examples of discrete random variables: number of arrivals in one second, number of attempts until success q A continuous-valued random variable takes on a range of real values, e. g. X(s) ranges from 0 to as s varies. q Examples of continuous(-valued) random variables: time when a particular arrival occurs, time between consecutive arrivals. q Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 24
Continuous-valued Random Variables q Thus, for a continuous random variable X, we can define its probability density function (pdf) q Note that since have is non-decreasing in x we for all x. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 25
Properties of Continuous Random Variables q From the Fundamental Theorem of Calculus, we have q In particular, q More generally, Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 26
Expectation of a Continuous Random Variable q The expectation (average) of a continuous random variable X is given by q Note that this is just the continuous equivalent of the discrete expectation Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 27
Important (Discrete) Random Variable: Bernoulli q The simplest possible measurement on an experiment: q Success (X = 1) or failure (X = 0). q Usual notation: q E(X)= Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 28
Important (discrete) Random Variables: Binomial q Let X = the number of success in n independent Bernoulli experiments ( or trials). P(X=0) = P(X=1) = P(X=2)= • In general, P(X = x) = Binomial Variables are useful for proportions (of successes. Failures) for a small number of repeated experiments. For larger number (n), under certain conditions (p is small), Poisson distribution is used. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 29
Binomial can be skewed or normal Depends upon p and n ! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 30
Important Random Variable: Poisson q A Poisson random variable X is defined by its PMF: Where q > 0 is a constant Exercise: Show that and E(X) = q Poisson random variables are good for counting frequency of occurrence: like the number of customers that arrive to a bank in one hour, or the number of packets that arrive to a router in one second. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 31
Important Continuous Random Variable: Exponential Used to represent time, e. g. until the next arrival q Has PDF q for some q Properties: q Need >0 to use integration by Parts! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 32
Memoryless Property of the Exponential q q An exponential random variable X has the property that “the future is independent of the part”, i. e. the fact that it hasn’t happened yet, tells us nothing about how much longer it will take. In math terms Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 33
Important Random Variables: Normal Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 34
Normal Distribution: PDF & CDF z q PDF: With the transformation: (a. k. a. unit normal deviate) q z-normal-PDF: q Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 35
Why is Gaussian Important? Uniform distribution looks nothing like bell shaped (gaussian)! Large spread ( )! CENTRAL LIMIT TENDENCY! Sample mean of uniform distribution (a. k. a sampling distribution), after very few samples looks remarkably gaussian, with decreasing ! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 36
Other interesting facts about Gaussian q Uncorrelated r. vs. + gaussian => INDEPENDENT! q Important in random processes (I. e. sequences of random variables) q Random variables that are independent, and have exactly the same distribution are called IID (independent & identically distributed) q IID and normal with zero mean and variance 2 => IIDN(0, 2 ) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 37
Height & Spread of Gaussian Can Vary! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 38
Rapidly Dropping Tail Probability! Sample mean is a gaussian r. v. , with x = & s = /(n)0. 5 ÞWith larger number of samples, avg of sample means is an excellent estimate of true mean. ÞIf (original) is known, invalid mean estimates can Shivkumar Kalyanaraman be rejected with HIGH confidence! Rensselaer Polytechnic Institute 39
Confidence Interval q Probability that a measurement will fall within a closed interval [a, b]: (mathworld definition…) = (1 - ) q Jain: the interval [a, b] = “confidence interval”; the probability level, 100(1 - )= “confidence level”; q = “significance level” q q Sampling distribution for means leads to high confidence levels, I. e. small confidence intervals Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 40
Meaning of Confidence Interval Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 41
Statistical Inference: Is A = B ? • Note: sample mean y. A is not A, but its estimate! • Is this difference statistically significant? • Is the null hypothesis y. A = y. B false ? Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 42
Step 1: Plot the samples Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 43
Compare to (external) reference distribution (if available) Since 1. 30 is at the tail of the reference distribution, the difference between means is NOT statistically significant! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 44
Random Sampling Assumption! Under random sampling assumption, and the null hypothesis of y. A = y. B, we can view the 20 samples from a common population & construct a reference distributions from the samples itself ! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 45
t-distribution: Create a Reference Distribution from the Samples Itself! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 46
t-distribution Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 47
Statistical Significance with Various Inference Techniques Normal population assumption not required t-distribution an approx. for gaussian! Random sampling assumption required Std. dev. estimated from samples itself! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 48
Normal, 2 & t-distributions: Useful for Statistical Inference Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 49
Relationship between Confidence Intervals and Comparisons of Means Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 50
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