Scientific Measurement

Transcription

1 Scientific Measurement SPA-4103 Dr Alston J Misquitta Lecture 5 - The Binomial Probability Distribution

2 Binomial Distribution Probability of k successes in n trials. Gaussian/Normal Distribution Poisson Distribution Probability of k events in an interval 2

3 Binomial Distribution Probability of k successes in n trials. Probability of k heads in n tosses with probability p of a head in a single toss. 3

4 l Consider 6 tosses of a coin (equivalent to single toss of 6 coins) l How often does this result in 6 heads (6h)? l P(6h) = (½) 6 = 1/64 = 1.6% l Repeat this experiment 100 times: we expect 6h to happen 1 or 2 times (maybe 0 or 3 times) l What is probability of 1 head & 5 tails i.e. P(1h)? (½) 1 x (½) 5? NO! This is P(httttt) what about P(thtttt) etc? P(1h) = 6(½) 1 x (½) 5 allows h on any of the 6 tosses P(1h) = 6/64 = 9.4% Quite likely to occur in 100 identical experiments 4

5 Binomial Distribution l What about P(4h)? P(4h) = (½) 4 x (½) 2 x N N = Number of combinations of getting 4 out of 6 No. of h N h h h h t t h h h t h t h h t h h t... h h h t t t h h t h t t h t h h t t... 4 h is like 2 h etc... Number of combinations of k from n 5

6 Binomial Distribution Number of ways of picking 4 heads from 6 tosses is: Here we took P(h) = P(t) = ½ i.e. P(success) = P(fail) What if P(success) P(fail)? If p = P(success) Binomial Distribution 6

7 Binomial Distribution Probability of k successes in n trials. Probability of k heads in n tosses with probability p of a head in a single toss. 7

8 Sir Francis Galton 8

9 Museum of Science, Boston 9

10 Binomial Distribution Probability of k successes in n trials. This is a discrete distribution. It describes distributions of true/false outcomes. The average value of k in n measurements (tosses of a coin; 1 for heads and 0 for tails) is: And standard deviation is 10

11 Binomial Distribution l In general p 0.5 e.g. Probability England winning next match. l Binomial statistics used for true/false type experiments l Example: opinion polls Ask a sample of people a question. Allow only yes/no answers Would you prefer to see the re-introduction of the death penalty to the UK? Poll 1000 randomly selected people n = 1000 Yes = 500 No (or Don't know ) = 500 Thus p = 500/1000 = 0.5 (50%!) What is the uncertainty? 11

12 Binomial Distribution l To reduce this error by factor of three to 1% we need to work very hard Need to increase n by factor of 10 for a factor 10 reduction in error! 12

13 Binomial Distribution Probability of k successes in n trials. Gaussian/Normal Distribution Poisson Distribution Probability of k events in an interval 13

14 Binomial Distribution Binomial distribution for various n values with p=0.4 Note the distribution is discrete - lines between points are to guide the eye f(k;n,p) f(k;n,p) is asymmetric around the peak Asymmetry reduces as n increases k 14

15 Binomial distribution to Gaussian/Normal distribution 15

16 Central Limit Theorem Why is the Gaussian Distribution so important? The Central Limit Theorem The mean of a large number of independent samples each of size n from the same probability distribution (not necessarily Gaussian) has approximately a Gaussian distribution centred on the population mean and variance which reduces as 1/n 16

17 Restatement of the Central Limit Theorem Form 2: If are independent random variables each drawn from the same distribution with expectation value and standard deviation, then the sample mean has an approximately normal distribution with expected value deviation when is sufficiently large. and standard True no matter what form the distribution of the random variables takes. How large n must be depends on the underlying distribution of. The random variables must be independent! Theorem can be generalised to the case where the random variables exhibit different distributions. 17

18 Illustration of the Central Limit Theorem See how the probablity distribution functions tend to the normal distribution as the number of dice throws increases. Here we have rolled n dice many times (till the distribution converges!) and have plotted the average of the values on the n dice. Wikipedia 18

19 Why is this relevant? many measurements table true length average Return to the measurement of our table Repeat this 100 times We will see a spread of measurements Spread arises from many small random effects Central Limit Theorem tells us that the spread will be Gaussian This explains why Gaussian errors appear everywhere! Dr Eram Rizvi Scientific Measurement - Lecture 6 19

20 The normal distribution arises no matter what the starting probabilities are! 5.5 Graphical illustration of the central limit theorem 165 p 1 = 1 / 6 p 2 = 1 / 6 p 3 = 1 / 6 p 4 = 1 / 6 p 5 = 1 / 6 p 6 = 1 / 6 Understanding Probability Henk Tijms (2ed, Cambridge) n=5 n=10 n=15 n=20 Fig Probability histogram for the unbiased die. 20

21 The normal distribution arises no matter what the starting probabilities are! 166 Probability and statistics Understanding Probability Henk Tijms (2ed, Cambridge) p 1 =0.2 p 2 =0.1 p 3 =0.0 p 4 =0.0 p 5 =0.3 p 6 =0.4 n=5 n=10 It takes more rolls of the biased die for a normal distribution to arise, but arise it does! n=15 n=20 Fig Probability histogram for a biased die. 21

22 Gaussian/Normal Distribution 22

23 Binomial Distribution Probability of k successes in n trials. Gaussian/Normal Distribution Poisson Distribution Probability of k events in an interval 23

24 Binomial distribution to Poisson distribution Discrete distribution k is a positive integer λ is the average number of events in an interval Describes probability of events in a period. (radioactive decay) Siméon Denis Poisson 24

25 1837 : Siméon-Denis Poisson ( ) published his Recherches sur la Probabilité des Jugements en Matière Criminelle et en Matière Civile : L. von Bortkiewicz explained the importance of the distribution in his book Das Gesetz der Kleinen Zahlen (The laws of small numbers). Example from book: Number of Prussian cavalry deaths attributed to fatal horse kicks in each of the years between 1875 and Here we have a large number of trials (cavalry men) and a small probability of success (a fatal horse kick). 25

26 Poisson Distribution Relevant for counting experiments Describes statistics of events occurring at a random but at a definite rate e.g. number of radioactive decays in 1 min interval number of births in UK per 24 hours Experiment: count number of radioactive decays in 1 min interval: k Assume 100% efficient detector - no error on k Repeat experiment get different k statistical fluctuation! n nuclei p prob of single decay in 1 min k decays measured in 1 min n ~ p ~ n p ~ finite As n and p 0 with np λ (const) then binomial dist gives Poisson dist. Probability of observing k counts if mean number of counts is λ λ = n x p = mean no. of counts k = observed no. of counts 26

27 Expectation value of a random variable X that is Poisson distributed: Therefore, to define a Poisson distribution all we need do is find the expectation value! 27

28 Important properties of the Poisson distribution In other words: measurement of k gives estimate of λ measurement of k gives estimate of σ i.e. the true mean is given by the mean of k and also k gives the true variance σ 2 Poisson dist. skewed for small λ More symmetric as λ increases One measurement gives λ and σ Thus error decreases as k increases but only as the square root! So to get an accurate radioactive decay count reading we need only one measurement, but it needs to be a very long one! 28

29 Binomial Distribution Probability of k successes in n trials. Gaussian/Normal Distribution Poisson Distribution Probability of k events in an interval 29