Chapter 6 Approaches to Assigning Probabilities Classical approach Relative frequency Subjective approach based on equally likely events assigning probabilities based on experimentation or historical data assigning probabilities based on the assignor s judgment ex weather forecast Complement of an event 1 P event occurring Joint probability intersection of events A and B Union of two events event containing all sample points in A or B or both Mutually Exclusive Events two events cannot occur together joint probability is 0 no points in common a 2 b 5 Marginal probabilities computed by adding across rows and down columns calculated in the margins Conditional probabilities used to determine how two events are related P A B A1 A2 P B P A B P B B1 B2 P A 1 00 Independence P A B P A Multiplication Rule P A B P A B P A Addition Rule P A or B P A P B P A and B Chapter 7 Discrete Random Variable one that takes on a countable number of values Continuous Random Variable one whose values are not countable Ex time 30 1 minutes 30 10000001 minutes etc Discrete Probability represent a population so use describe them using various parameters E X xP x weighted average population mean V X 2 OR 2 x2 P x 2 variance 2 standard deviation Laws of ExpectedValue E c c E X c E x c E cX cE X Bivariate Distributions Covariance COV X Y x y xyP x y x y Coefficient of Correlation COV X Y x y Laws of Variance V c 0 V X c V X V cX c2V X Laws E X Y E X E Y V X Y V X V Y 2COV X Y If X and Y are independent COV X Y 0 and thus V X Y V X V Y Binomial Random Variable counts the number of successes in n trials of the binomial experiment Poisson Experiment discrete probability distribution that refers to the number of events successes within a specific time period or region of space number of flaws in cloth number of accidents in one day on a particular highway etc P X n x n x p x Binomial Distribution np 2 np 1 p np 1 p P x e x x Chapter 8 Continuous Random Variables Probability Density Functions f x 1 b a where a x b example P 2500 x 3000 3000 2500 x 1 b a Normal Distribution bell shaped and symmetrical around the mean Z X mean standard deviation Exponential Distribution e x x 0 Student t distribution v nu degrees of freedom and gamma is k value Chi square distribution is not symmetrical F Distribution density function which has 2 v s and degrees of freedom in numerator and denominator Chapter 9 Sampling Distribution distribution created by sampling Mean Variance 2 n Standard deviation n Z xbar n Central Limit Theorem Sampling distribution of the mean of a random sample drawn from any population is approximately normal for sufficiently large sample size 30 If the population is normal then xbar is normally distributed for all values of n If it is non normal then it is approximately normal for only larger values of n Example P Xbar 32 P xbar expected mean standard deviation P Z Sampling Distribution of a Proportion Count the number of successes in a sample and compute Phat X n p 1 p n Square root of whole thing np Phat is the standard error of proportion