Probability & Statistical Inference Lecture 3

1. Probability & Statistical Inference Lecture 3 MSc in Computing (Data Analytics)

2. Lecture Outline • A quick recap • Continuous distributions. • Question Time

3. A Quick Recap

4. Probability & Statistics • We want to make decisions based on evidence from a sample i.e. extrapolate from sample evidence to a general population • To make such decisions we need to be able to quantify our (un)certainty about how good or bad our sample information is. MakeInference Describe

5. Some Definitions • An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. • The set of all possible outcomes of a random experiment is called the sample space of an experiment and is denote by S • A sample space is discrete if it consists of a finite or countable infinite set if outcomes. • A sample space is continuous if it contains an interval or real numbers. • An event is a subset of the sample space of a random experiment.

6. Some Definitions • A sample space is discrete if it consists of a finite or countable infinite set if outcomes. • A sample space is continuous if it contains an interval or real numbers. • An event is a subset of the sample space of a random experiment.

7. Probability • Whenever a sample space consists of n possible outcomes that are equally likely, the probability of the outcome 1/n. • For a discrete sample space, the probability of an event E, denoted by P(E), equals the sum of the probabilities of the outcome in E. • Some rules for probabilities: • For a given sample space containing n events E1, E2, E3, ........,En • All simple event probabilities must lie between 0 and 1: 0 <= P(Ei) <= 1 for i=1,2,........,n • The sum of the probabilities of all the simple events within a sample space must be equal to 1:

8. Discrete Random Variable • A Random Variable (RV) is obtained by assigning a numerical value to each outcome of a particular experiment. • Probability Distribution: A table or formula that specifies the probability of each possible value for the Discrete Random Variable (DRV) • DRV: a RV that takes a whole number value only

9. Summary Continued… • For Discrete RV we often have a mathematical formula which is used to calculate probabilities, i.e. P(x) = some formula • This formula is called the Probability Mass Function (PMF) • Given the PMF you can calculate the mean and variance by: • When the summation is over all possible values of x

10. Binomial Distribution – General Formula • This all leads to a very general rule for calculating binomial probabilities: In General Binomial (n,p) n = no. of trials p = probability of a success x = RV (no. of successes) • Where P(X=x) is read as the probability of seeing x successes.

11. Binomial Distribution • If X is a binomial random variable with the paramerters p and n then

12. Poisson Probability Distribution • Probability Distribution for Poisson Where  is the known mean: • x is the value of the RV with possible values 0,1,2,3,…. e = irrational constant (like ) with value 2.71828… • The standard deviation , , is given by the simple relationship;  =

13. Continuous Probability Distributions

14. Continuous Probability Distributions • Experiments can lead to continuous responses i.e. values that do not have to be whole numbers. For example: height could be 1.54 meters etc. • In such cases the sample space is best viewed as a histogram of responses. • The Shape of the histogram of such responses tells us what continuous distribution is appropriate – there are many.

15. Normal Distribution (AKA Gaussian) • The Histogram below is symmetric & 'bell shaped' • This is characteristic of the Normal Distribution • We can model the shape of such a distribution (i.e. the histogram) by a Curve

16. Normal Distribution • The Curve may not fit the histogram 'perfectly' - but should be very close • Normal Distribution - two parameters, µ = mean,  = standard deviation, • The mathematical formula that gives a bell shaped symmetric curve f(x) = Height of curve at x =

17. Normal Distribution • Why Not P(x) as before? => because response is continuous • What is the probability that a person sampled at random is 6 foot? • Equivalent question: what proportion of people are 6 foot? • => really mean what proportion are • 'around 6 foot' ( as good as the measurement device allows) - so not really one value, but many values close together.

18. Example: What proportion of graduates earn €35,000? • Would we exclude €35,000.01 or €34,999.99? • Round to the nearest €, €10, €100, €1000? • Continuous measure => more useful to get proportion from €35,000 - €40,000 • Some Mathematical Jargon: • The formula for the normal distribution is formally called the normal probability density function (pdf)

19. Can visualise this using the histogram of salaries. The Shaded portion of the Histogram is the Proportion of interest

20. Since the histogram of salaries is symmetric and bell shaped, we model this in statistics with a Normal distribution curve. Proportion = the proportion of the area of the curve that is shaded • So proportions • = proportional area under the curve • = a probability of interest • Need; • To know ,  • To be able to find area under curve

21. Area under a curve is found using integration in mathematics. • In this case would need a technique called numerical integration. • Total area under curve is 1. • However, the values we need are in Normal Probability Tables.

22. Standard Normal • The Tables are for a Normal Distribution with •  = 0 and  = 1 • this is called the Standard Normal • Can 'convert' a value from any normal to the standard normal using standard scores (Z scores) Corresponding Value from Normal  = 0  = 1 Value from any Normal Distribution Standardize

23. Z-Score Example Z scores are a unit-less quantity, measuring how far above/below  a certain score (x) is, in standard deviation units. Example: A score of 35, from a normal distribution with  = 25 and  = 5. Z = ( 35 − 25) / 5 => 10/5 = 2 So 35 is 2 standard deviation units above the mean What about a score of 20 ? Z = ( 20 - 25) / 5 => − 5 / 5 = − 1 So 20 is 1 standard unit below the mean

24. Z-Score Example Positive Z score => score is above the mean Negative Z score => score is below the mean By subtracting  and dividing by the  we convert any normal to  = 0, =1, so only need one set of tables!

25. Example: From looking at the histogram of peoples weekly receipts, a supermarket knows that the amount people spend on shopping per week is normally distributed with:  = €58  = €15.

26. What is the probability that a customer sampled at random will spend less than €83.50 ? Z = ( x−  ) /  = ( €83.50 - €58 ) / €15 => 1.7 Area from Z=1.7 to the left can be read in tables From tables area less than Z = 1.7 => 0.9554 So probability is 0.9554 Or 95.54%

28. What is the probability that a customer sampled at random will spend more than €83.50 ? Z = ( x−  ) /  = ( €83.50 - €58 ) / €15 => 1.7 Area from Z=1.7 to the right can be read in tables From tables area greater than Z = 1.7 => 1- 0.9554 = 0.0446 So probability is 0.0446 Or 4.46%

29. Exercise • Find the proportion of people who spend more than €76.75 • Find the proportion of people who spend less than €63.50 • Note: The tables can also be used to find other areas (less than a particular value, or the area between two points)

30. Characteristics of Normal Distributions • Standard Deviation has particular relevance to Normal distribution • Normal Distribution => Empirical Rule

31. The normal distribution is just one of the known continuous probability distributions. • Each have their own probability density function, giving different shaped curves. • In each case, we find probabilities by calculating areas under these curves using integration. • However, the Normal is the most important – as it plays a major role in Sampling Theory.

32. Other important continuous probability distributions include • Exponential distribution – especially positively skewed lifetime data. • Uniform distribution. • Weibull – especially for ‘time to event’ analysis. • Gamma distribution – waiting times between Poisson events in time etc. • Many others…..

33. Summary – Random Variables • There are two types – discrete RVs and continuous RVs • For both cases we can calculate a mean (μ) and standard deviation (σ) • μ can be interpreted as average value of the RV • σ can be interpreted as the standard deviation of the RV

34. Summary Continued… • For Discrete RV we often have a mathematical formula which is used to calculate probabilities, i.e. P(x) = some formula • This formula is called the Probability Mass Function (PMF) • Given the PMF you can calculate the mean and variance by: • When the summation is over all possible values of x

35. Summary Continued… • For continuous RVs, we use a Probability Density Function (PDF) to define a curve over the histogram of the values of the random variables. • We integrate this PDF to find areas which are equal to probabilities of interest. • Given the PDF you can calculate the mean and variance by: • Where f(x) is usual mathematical notation for the PDF

36. Question Time

37. Next Week • Next week we will start with the practical part of the course. We will move to Lab 1005 in Aungier Street