Ch5. Probability Densities II Dr. Deshi Ye

2/ Other Prob. Distribution  Uniform distribution: equally likely outcome Mean of uniform Variance of uniform

3/33 Ex.  Students believe that they will get the final scores between 80 and 100. Suppose that the final scores given by the instructors has a uniform distribution.  What is the probability that one student get the final score no less than 85?

4/33 Solution  P(85  x  100)= (Base)(Height)  = ( )(0.05) = f(x)f(x)f(x)f(x) x

5/ The Log-Normal Distr.  Log-Normal distribution: It has a long right-hand tail By letting y=lnx Hence

6/33 Mean of Log-Normal  Mean and variance are Proof.

7/33 Gamma distribution Mean and Variance

8/33 The Exponential Distribution  By letting in the Gamma distribution Mean and Variance

9/ The Beta Distribution  When a random variables takes on values on the interval [0,1] Mean and Variance

10/33 Beta distribution  Are used extensively in Bayesian statistics  Model events which constrained to take place within a interval defined by minimum and maximum value  Extensively used in PERT, CPM, project management

11/ Weibull Distribution Mean and Variance

12/33 Weibull distribution  Is most commonly used in life data analysis  Manufactoring and delivery times in industrial engineering  Fading channel modeling in wireless communication

13/ Joint distribution  Experiments are conduced where two or more random variables are observed simultaneously in order to determine not only their individual behavior but also the degree of relationship between them.

14/33 Two discrete random variables The probability that X 1 takes value x 1 and X 2 will take the value x 2 EX. x x

15/33 Marginal probability distributions x x EX.

16/33 Conditional Probability distribution The conditional probability of X 1 given that X 2 = x 2 If two random variables are independent

17/33 EX.  With reference to the previous example, find the conditional probability distribution of X 1, given that X 2 =1. Are X 1 and X 2 independent?  Solution. Hence, it is dependent

18/33 Continuous variables  If are k continuous random variables, we refer to as the joint probability density of these random variables

19/33 EX.  P179. Find the probability that the first random variable between 1 and 2 and the second random variable between 2 and 3

20/33 Marginal density  Marginal density of X 1 Example of previous

21/33 Distribution function

22/33 Independent If two random variables are independent iff the following equation satisfies.

23/33 Properties of Expectation  Consider a function g(x) of a single random variable X. For example: g(x) =9x/  If X has probability density f(x), then the mean or expectation of g(x) is given by Or

24/33 Properties of Expectation If a and b are constants Proof. Both in continuous and discrete case

25/33 Covariance Covariance of X 1 and X 2 : to measure Theorem. When X 1 and X 2 are independent, their covariance is 0

26/ Checking Normal  Question: A data set appears to be generated by a normal distributed random variable  Collect data from students’ last 4 numbers of mobiles

27/33 Simple approach  Histogram can be checked for lack of symmetry  A single long tail certainly contradict the assumption of a normal distribution

28/33 Normal scores plot  Also called Q-Q plot, normal quantile plot, normal order plot, or rankit plot. Normal scores: an idealized sample from the standard normal distribution. It consists of the values of z that divide the axes into equal probability intervals. For example, n=4.

29/33 Steps to construct normal score plot  1) order the data from smallest to largest  2) Obtain the normal scores  3) Plot the i-th largest observation, versus i-th normal score mi, for all i. Plot

30/33 Normal scores in Minitab  In minitab, the normal scores are calculated in different ways: The i-the normal score is Where is the inverse cumulative distribution function of the standard normal

31/33 Property of Q-Q plot  If the data set is assumed to be normal distribution, then normal score plot will resemble to a /line through the original.

32/ Transform observation to near normality  When the histogram or normal scores plot indicate that the assumption of a normal distribution is invalid, transformations of the data can often improve the agreement with normality. Make larger values smaller Make large value larger

33/33 Simulation  Suppose we need to simulate values from the normal distribution with a specified From The value x can be calculated from the value of a standard normal variable z 1)z can be obtained from the value for a uniform variable u by numerically solving u=F(z) 2)Box-Muller-Marsaglia method: it starts with a pair of independent variable u 1 and u 2, and produces two standard normal variables

34/33 Box-Muller-Marsaglia Then It starts with a pair of independent variable u 1 and u 2, and produces two standard normal variables

35/33 Simulation from exponential distribution  Suppose we wish to simulate an observation from the exponential distribution The computer would first produce the value u from the uniform distribution. Then

36/33 Population and sample

37/33 Population and Sample  Investigating: a physical phenomenon, production process, or manufactured unit, share some common characteristics.  Relevant data must be collected.  Unit: the source of each measurement. A single entity, usually an object or person  Population: entire collection of units.

38/33 Population and sample Population sample

39/33 Key terms  Population All items of interest  Sample Portion of population  Parameter Summary Measure about Population  Statistic Summary Measure about sample

40/33 Examples PopulationUnitvariables All students currently enrolled in school studentGPA Number of credits All books in library bookReplacement cost

41/33 Sample  Statistical population: the set of all measurement corresponding to each unit in the entire population of units about which information is sought.  Sample: A sample from a statistical population is the subset of measurements that are actually collected in the course of investigation.

42/33 Sample  Need to be representative of the population  To be large enough to contain sufficient information to answer the question about the population

43/33 Discussion  P10, Review Exercises 1.2  A radio-show host announced that she wanted to know which singer was the favorite among college students in your school. Listeners were asked to call and name their favorite singer. Identify the population, in terms of preferences, and the sample.  Is the sample likely to be more representative?  Comment. Also describe how to obtain a sample that is likely to be more representative.