Random variables (section 6.1)

Transcription

1 Random variables (section 6.1) random variable: a number attached to the outcome of a random process (r.v is usually denoted with upper case letter, such as \) discrete random variables discrete random variable: r.v. with finite (or countably infinite) number of possible values ex. flip coin 4 times, count number of heads \œ# heads in 4 trials (before the experiment is done!) B : an outcome of \ (possible values 0, 1, 2, 3, 4) probability distribution of \: all the possible outcomes of \ and their probabilities outcome B P \œb HTTH HTHT HTTT HHTT HHHT THTT THTH HHTH TTHT THHT HTHH TTTT TTTH TTHH THHH HHHH

2 ex. draw a household at random in the US; inhabitants count the # of \œ# inhabitants in a randomly drawn household possible values ( B): 1, 2, 3, 4, 5, 6, 7, 8 Ÿ outcome B Ÿ P \œb negligible ex. games of chance, with betting \œnet gain/losses (possibly negative!) in a lottery: pay $1 to play pick a number between 000 and 999 inclusive 3-digit number then drawn at random win $500 if your number is drawn outcome B P \œb

3 probability histogram: areas (or heights) of rectangles are probabilities ex. probability histograms for the Poisson distribution, which is the distribution of the number of particle decays per unit time given off by some radioactive substance (each substance has its own half life ) here lambda is log/ 2 Î half life

4 continuous random variables continuous random variable: value of the r.v. is a real number (uncountably infinite number of possible values in an interval of the real line) ex. choose a real number at random between zero and one (spinner), call it \ \ is said to have a uniform distribution density curve for a continuous random variable: the area between + and, under a density curve gives the probability that \ will be in the interval between + and,.

5 ex. density curve of a uniform distribution, along with relative frequency histogram of 30,000 computer-simulated uniform random numbers:

6 for a continuous ramdom variable \, area under the density curve between + and, is P + Ÿ \ Ÿ, ex. for the uniform distribution, area œ, + P.2 Ÿ \ Ÿ.4 œ.4.2 œ.2 density curves in general: P +Ÿ\Ÿ, œ area under the density curve between + and,

7 notes on density curves area under density curve is 1 idealized relative frequency histogram used as models of the relative frequencies of observations in a population

8 Mean (or expected value) of a random variable \ discrete random variable: mean is weighted average of possible outcomes weights are the probabilities of the outcomes mean is a measure of the center of the probability histogram. œ B P \ œ B ( œ BP B ) B B ex. \œ # heads in 4 coin tosses mean of \ œ œ! œ œ œ 2

9 ex. \œ # inhabitants in a randomly drawn household. œ ( negligible) œ 2.6 ex. \œ net gains/losses in lottery. œ œ œ 0.5 ($) ex. A binary (or Bernoulli) random variable \ can be defined from a categorical variable with two categories: E : event that voter is Republican let \ œ 1 E - : event that voter is not Republican let \ œ 0 Suppose P E œ P \ œ 1 œ : outcome B 0 1 P \œb 1 : : mean:. œ 0 1 : 1 : œ :

10 ex. baseball strategy: each situation in an inning gives a discrete probability distribution for the number of runs scored in the remainder of the inning (G. Lindsey, 1959 & 1960 seasons) bases runs scored in the remainder of the inning occupied outs Ÿ none expected number of runs œ â 3 0 œ probability of scoring 1 or more runs 1,2 0 œ œ , , full 0 1 2

11 expected number of runs in the remainder of the inning bases occupied outs ,2 1,3 2,3 full probability of scoring one or more runs in remainder of inning bases occupied outs ,2 1,3 2,3 full

12 continuous random variable: the density curve (calculus) mean is a balance point of

13 The normal distribution (section 6.2) Density curve: used as a mathematical model of a frequency distribution for a population of measurements ( mathematical relative frequency histogram ) 1. mathematical function 2. non-negative 3. area under it is 1. ex. area between + and, is the proportion of observations having values between + and, median divides area in half; mean is the balance point Normal curve (bell-shaped curve) model that describes many situations fairly well (due to the central limit theorem) symmetric around the mean. (. is also the median) standard deviation 5 measures the spread

14 normal curves with same 5, different.'s: normal density curve normal curves with same., different 5's: x normal density curve x

15 rule of thumb: area between. 5 and. 5 is.68 (approx.) area between. 2 5 and. 2 5 is.95 (approx.) area between. 3 5 and. 3 5 is.997 (approx.)

16 Standard normal distribution:. œ 0, 5 œ 1 ( œ 5 # ) Areas under any normal distribution curve can be found by transforming to the standard normal distribution. area to the left of B in a normal(., 5) distribution: find standardized value of B: Dœ B. 5 this is called the D score (represents value of B re-expressed in units of standard deviations above or below the mean) look up area to the left of D in the table (text, Appendix A pages A1-A2) of the standard normal distribution:

17 scores on SAT-MATH:. is about 500; 5 is about 100 ex. what proportion of math scores are below 650? (What percentile is the score 650?) want area to the left of 650 ( œb): Dœ B œ œ (a score of 650 is 1.5 standard deviations above the mean) from table, area to the left of Dœ1.50 is.9332 An SAT score of 650 is the 93rd percentile. ex. What proportion of SAT scores are above 575? Dœ œ.75

18 area to the left of D is area œ œ.2266 ex. What proportion of scores are between 450 and 525? " 100 " D œ œ.50 area œ # 100 # D œ œ.25 area œ.5987 area# area " œ œ.2902

19 ex. What SAT score represents the 90th percentile? reverse table look-up: find the corresponds to an area of.90 D value that area œ.8997 (closest in the table); D œ 1.28 the 90th percentile is approximately 1.28 standard deviations above average; now convert this D score to the original scale: Dœ B. 5 Ê B œ 5 D. Bœ100(1.28) 500 œ628 is the 90th percentile

20 A few useful percentiles of the normal distribution: area to the left D CI % coverage

21 Recently discovered formula for approximating the cumulative area in a standard normal distribution: area to the left of D 1 usually good to 3 decimal places! 1 exp D$ D (Bowling, S. R. et al Journal of Industrial Engineering and Management 2: )

22 Normal quantile plot: graphical method of assessing whether data arose from a normal distribution idea: suppose an observation is the third smallest in a data set with 80 observations. Calculate what we would expect the third smallest observation to be if the data were normally distributed, & compare with the actual. Do this with all the observations & plot actual vs ideal normal ordered rank obs percentile corresponding 3 B areas D ã ã ã ã ã plot B3's vs. D3's (or vice versa): points should resemble a straight line

23 80 observations simulated in MINITAB from a normal distribution with. œ 500 and 5 œ 100 Probability Plot of sat Normal - 95% CI Percent Mean StDev N 80 AD P-Value sat

24 80 observations simulated in MINITAB from a rightskewed distribution (a gamma distribution ) with a mean of 10 Probability Plot of gamma Normal - 95% CI Percent Mean StDev N 80 AD P-Value < gamma