Statistics Lecture 6

zToday: Finished 4.5 ; began discrete random variables ( ) zToday: Finish discrete random variables ( ) and begin continuous R.V.’s ( ) zNext Day: 6.4, 6.6 and 7.1 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

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:

Binomial Distribution zBinomial distribution is a distribution that models chance variation of n repetitions of an experiment that has only 2 outcomes zRandom variable, X, is the count of observations falling in one of the categories zX is number of successes in n Bernoulli trials zX = number of successes zProbability of success remain constant

zn = Number of trials zp = probability of success zX = number of successes in the n trials (for X =0,1,2,…, n) z zNote:

Example 5.67 z15% of trees in a forest have leaf damage zIf 5 are selected at random, find probability: y3 have leaf damage yno more than 2 have leaf damage y3 do not have leaf damage

Mean and Standard Deviation zMean: zStandard Deviation:

Continuous Random Variables zCan you list all points in an interval zHave described distribution of quantitative variable using a histogram zA relative frequency histogram has proportions on Y-axis. zSum of bar heights is

zCan describe overall shape of distribution with a mathematical model called a density function, f(x) zDescribes main features of a distribution with a single expression zTotal area under curve is zArea under a density curve for a given range gives

zTotal area under curve is z

Features zMode: zMedian zQuartiles zMean

Standardizing zIf X is a R.V., the standardized variable, Z, has mean 0 and standard deviation 1.

Normal Distributions zCommon continuous density is the normal distribution zIt is symmetric, bell-shaped and uni-modal zDenoted

zWhat happens if mean is changed? zWhat happens if standard deviation is changed?

Relative Location of Mean and Median zRight Skewed zLeft Skewed zSymmetric