Statistics Lecture 5

zLast class: Finished and started 4.5 zToday: Finish 4.5 and begin discrete random variables ( ) zNext Day: More discrete random variables ( ) and begin continuous R.V.’s ( ) zAssignment #2: 4.14, 4.24, 4.41, 4.61, 4.79, 5.13(a and c), 5.32, 5.68, 5.80 zDue in class Tuesday, October 2

Independent Events zTwo events are independent if: zThe intuitive meaning is that the outcome of event B does not impact the probability of any outcome of event A zAlternate form:

Example zFlip a coin two times zS= zA={head observed on first toss} zB={head observed on second toss} zAre A and B independent?

Example zMendel used garden peas in experiments that showed inheritance occurs randomly zSeed color can be green or yellow z{G,G}=Green otherwise pea is yellow zSuppose each parent carries both the G and Y genes zM ={Male contributes G}; F ={Female contributes G} zAre M and F independent?

Example (Randomized Response Model) zCan design survey using conditional probability to help get honest answer for sensitive questions zWant to estimate the probability someone cheats on taxes zQuestionnaire: y1. Do you cheat on your taxes? y2. Is the second hand on the clock between 12 and 3? y YES NO zMethodology: Sit alone, flip a coin and if the outcome is heads answer question 1 otherwise answer question 2

More on Probability zWill take a more formal look at describing random phenomenon zA random variable, X, associates a numerical value to each outcome of an experiment zWill consider two types: yDiscrete random variables yContinuous random variables

Discrete versus Continuous zDiscrete random variables have either a finite number of values or infinitely many values that can be ordered in a sequence zContinuous random variables take on all values in some interval(s)

Examples zDiscrete or continuous yNumber of people arriving in a supermarket yHair color of randomly selected people yWeight lost from a diet program yRandom number between 0 and 4

Discrete Random Variables zDescribe chances of observing values for a discrete random variable by probability distribution zProbability distribution of a discrete random variable, X, is the list of distinct numerical outcomes and associated probabilities zIf distribution is estimated from data, it is called the empirical distribution

Properties z for each value x i of X z

zCan display distribution using a probability histogram zX-axis represents outcomes zY-axis is the probability of each outcome zUse rectangles, centered at each value of X, to display probabilities

Example zProbability distribution for number people in a randomly selected household zDraw the probability histogram

Mean and Variance for Discrete Random Variables zSuppose have 1000 people in a population (500 male and 500 female) and average age of the males is 26 and average age of females is 24 zWhat is the mean age in the population? zSuppose have 1000 people in a population (900 male and 100 female) and average age of males is 26 and average age of females is 24 zWhat is the mean age in the population?

zMean must consider chance of each outcome zMean is not necessarily one of the possible outcomes zIs a weighted average of the outcomes

zThe mean (or expected value) of a discrete R.V., X, is denoted E(X) zIs also denoted as z

zVariance of a discrete R.V. weights the squared deviations from the mean by the probabilities zThe standard deviation is

Example zCompute mean and variance of number of people in a household

Example (true story) zPeople use expectation in real life zParking at Simon Fraser University (B.C., Canada) is $9.00 per day zFine for parking illegally is $10.00 zWhen parking illegally, get caught roughly half the time zShould you pay the $9.00 or risk getting caught?

zProbability Model - is an assumed form of a distribution of a random variable

Bernoulli Distribution zBernoulli distribution: yEach trial has 2 outcomes (success or failure) yProb. of a success is same for each trial yProb. of a success is denoted as p yProb. of a failure, q, is yTrials are independent zIf X is a Bernoulli random variable, its distribution is described by zwhere X=0 (failure) or X=1 (success)

Example zA backpacker has 3 emergency flares, each which light with probability of zFind probability the first flare used will light zFind probability that first 2 flares used both light zFind probability that exactly 2 flares light

Mean and Standard Deviation zMean: zStandard Deviation: