STAT1201 Module 4 Notes 4.1 Discrete Distributions Binomial probabilities Let X be the number of females in sample of three individuals x P(X = x) 0 0.125 1 0.375 2 0.375 3 0.125 Bernoulli trial Outcomes labelled success or failure Count of success in s series of Bernoulli trials with a constant probability is called binomial distribution For the female example above p is probability of female X is the success count n is number of trials X ~ Binomial (n,p) x P(X = x) 0 (1-p)3 1 3p(1 - p)2 2 3 3p2(1 - p) p3 The formula for the probability of a certain number of successes (P(X = x)) when X ~ Binomial (n,p) is P X= x)=| n p (1-p)"-x x Where n = n Cx X Is the binomial coefficient. Find on calculator by pressing SHIFT+DIVIDE Finds number of ways objects can be chosen E.g. 2 people sampled from a group of 10. There are: ( 10 =45 2 Different pairs of people that could be chosen IGNORES ORDER OF PICKS Binomial Tables for small p and n values Hypergeometric distribution - Calculate probabilities for finite populations - In a Bernoulli test the probability usually stays the same after each pick, independent, infinite or very large population - If population is small then the chance of E.g. picking a male student will decrease every time a male student is picked. Smaller the population the bigger the probability change.
4.1.2 Sample Counts and Proportions Expected value of X = np E(X)=np Variance (X) = Var (X)=np(1-p) Standard deviation (X) = Vnp(1-p) 4.1.3 Maximum Likelihood Estimation The expected sample proportion value gives E(?)=p Sample proportion an unbiased estimator of population proportion Maximum likelihood estimation Looks for the probability value p that gives the highest P(X = p) E.g. n=20 X=16 p (1-p)2º 20-16-4845 p16(1-p)4 16/ 20 , 16( So if the probability of success was 0.6 and there were 16 successes out of 20 then the probability of this P(X=16)=0.035 but if the probability of success was 0.75 then P(X=16)=0.130 . Which value of p makes it most likely to get 16 successes. Can solve this question using calculus But also with plot of the likelihood function, P(X=16| pi
4.2.1 Normal Density Curves Given by the probability density function 1 1(x-1) 12 f(x)= 1 2 ?? Where u and o are parameters of the function (mean and s.d.) This curve is called normal density curve - Normal curve symmetric around value x= u If X is continuous random variable, whose probabilities are given by normal density curve with parameters u and o then we say X has a Normal distribution and write X Normal(p,0) As a rough rule for Normal distributions - Area within 1 standard deviation of the mean is 68% - Area within 2 standard deviation of the mean is 95% - Area within 3 standard deviation of the mean is 99.7% R code: pnorm(c(1,2,3)) - pnorm(c(-1,-2,-3)) 4.2.2 Standard Units Standard Normal distribution Normal distribution standardised