STAT131 Week 6 L1b Introduction to Continuous Random Variables

1. STAT131 Week 6 L1b Introduction to Continuous Random Variables Anne Porter alp@uow.edu.au

2. Sample Population Model Discrete Random Variable Review: Sample and ModelCentre and spread Centre Spread Variance Spread Standard Deviation Sx2 Sx

3. Binomial~ (n,p) Population Model DR Variable Review Discrete Random Variables & Specific case of the Binomial Centre Spread Variance Model Fit Calibration Poisson( ) E(X)=np Var(X)=np(1-p)

4. Discrete Versus Continuous • Discrete measurement • Continuous There is no possible measure between the points on the scale Examples of discrete random variables: The number of successes in n fixed trials - Binomial The count of some event per unit time The set of possible values of the RV is neither finite not countably infinite.

5. Examples: Continuous RV • What is a sample space? Sample space is the values that the random variable can possibly take on. • What is the sample space that the life of a battery can take on? That is the battery does not ever work, 0 units of time, to possibly working for ever

6. Features of models • What features of models are we interested in? • Shape, probability of some event happening • Centre • Spread • Fit of data to a model • Calibration of the model Note that in some instances we don’t actually need to estimate spread eg Poisson where Mean = Variance

7. Probability for Continuous Random Variables • With our infinitely divisible measure of life time of the battery we are not interested in the probability of a precise individual outcome. • Example: P(X=5.3456743234456...units of time) • P(X=a)=0 for any constant a. The probability of any precise value is approximately zero. • The data are continuous, infinitely divisible Hence we are interested in intervals of values Example: P( 10 <X < 20)

8. Finding probability Is there any difference in answer for these • P( 10 <X < 20) • P( 10 <X < 20) • P( 10 <X < 20) • P( 10 < X < 20) No because the probability of exactly equalling a constant given a continuous random variable is zero

9. Finding probability: Continuous Random Variables • How can we find a estimate of the probability function of the continuous random variable, the lifetime of batteries? Test the lifetime of a number of batteries , that is, use the relative frequency approach.

10. Finding probability: Continuous Random Variables • How could we visually represent the relative frequency of the random variable? A histogram

11. As n increases What happens to the histogram as n increased • The Rel Freq(a<X<b)-----------> Prob(a<X<b) The curve becomes a smooth line Different random variables have different probabilities associated with their events and hence the different probability density functions have different shapes.

12. Finding probability: Continuous Random Variables • How do we find the area in the histogram? By summing the areas of all the individual bars • How do we find the area under a curve? By finding the area of the smallest possible bars and as the size decreases it becomes finding the integral over the domain

13. What were the conditions for defining a discrete random variable? • All px>0 and all px<1 • Sum of all px=1

14. Conditions defining a Continuous Random Variable • From the 'area =probability' property the following hold • f(x)> 0 for all and, • the total area is 1, ie. • the probability of a value between some interval a and b is given by

15. Cumulative probability function • Just as for a discrete random variable we could accumulate the probabilities to produce a cumulative probability function we can do the same for continuous random variables. It is the probability that X has a value less than or equal to x. This is expressed as F(x)=P(X<x) where thethe area to the left of x is given by • At x= the area is 0 and at x= the area is 1.

16. Cumulative probability function F(X)=1 F(X)=0

17. A comparison between discrete and continuous random variables. • Discrete random variables takes on a finite number or countably infinite number of values, whilst • Continuous random variablesconsists of one or more (possibly infinite) intervals on the real line

18. A comparison between discrete and continuous random variables. • For discrete random variables the probability of outcome x is given by P(X=x) or px • For a continuous r.v P(X=a)=0 for any constant a. Hence we are interested in intervals of values P(a<X<b), eg P( 10 <X < 20). • For continuous random variables probability is determined for an interval eg for outcome a to b

19. An example • For the random variable with the pdf defined as illustrated, y=1/2 0<x<2. Find the area under the function by using • Area=Length by Breadth • Integration Area=2x0.5=1 square unit

20. 1 An example • What is the probability of P(0<X<1)? 0.5 How did we do this? By use of symmetry

21. Centre • The sample mean can provide an estimate of the population mean, E(X). • In both discrete and continuous cases this is provided by

22. Population mean for Random Variables • The population mean symbolised is found for • discrete random variables by and for •  continuous random variables

23. An example • What mean for this distribution 1 How did we do this? By use of symmetry How do we calculate the mean For a discrete random variable? How then do we calculate the mean for a continuous random variable?

24. Spread • The sample variance is s2This can be used as an estimate of • For both discrete and continuous r.v this is

25. Spread • For discrete random variables this is given by or • For continuous random variables this is given by or

26. Continuous Random variable Histogram (observed & expected ) - The bins can have different widths - The bins can start in different positions - This will affect the numbers and the numbers expected in cells - The expected count in each > 5 • Discrete Random Variable • Bar Chart • observed & • expected counts Goodness of fit

27. Goodness of Fit • To see if the data observed fit a given model • Calculate • For discrete models d= • For continuous models d=(g-p-1) d=g-p-1 where p is the number of parameters estimated So for the Poisson () we have one parameter 

28. We will continue ... • Looking at two specific continuous distributions