Preparation for Midterm 2, Friday March 19. The midterm is over Devore 3.4 - 4.4, except: --in 3.5 we only cover the hypergeometric and geometric distributions only (not a general negative binomial distribution) --in 4.4 we only cover the exponential distribution (not the general gamma or chi-squared distributions) There will be no calculators. You will have to do some simple arithmetic. You will be given a copy of any table of values you need for probability distributions. If you don't have a table, put numbers into an appropriate formula. Binomial(n,p) distribution. Understand how Binomial(n,p) describes sampling with replacement. Know mean = np variance = np(1-p) standard deviation = square root of variance Know approximation of binomial(n,p) by Poisson(lambda), where lambda = np, when n is large and p small: rule of thumb: this approximation is ok for n at least 100, p at most .01, np at most 20. Know approximation by normal distribution for n large and p, (1-p) not too small: rule of thumb: this apporoximation is ok for both np and n(1-p) at least 10. Be able to use the continuity correction for your normal approximation. Hypergeometric distribution Understand meaning in terms of sampling without replacement. Know mean. Be able to write out the correct computation of probability if needed. Poisson(lambda) Distribution Know the definition, pmf, cdf. Be able to use tables if needed. Know mean (lambda) and variance (lambda). Know interpretation as giving the distribution of number of events in a time interval and be able to use this in problems. Know that if the Poisson distribution applies to events occuring at rate lambda per unit time, then the waiting time to the next event has distribution exponential(lambda). Exponential(lambda) distribution Know definition, pmf, cdf. Easiest to remember: for rv X with this distribution, for any t>0, Prob(X > t) = exp{-lambda t} . Note, this tells you Prob(X > t(1/lambda)) = exp{-t} which doesn't depend on lambda. You can think of this as saying that after rescaling time to multiples of the average waiting time, the exp. distributions are all the same. Think of X as the time you have to wait for the next event, where events occur at a steady rate of lambda events per unit time independent of previous events. Note, here: - your expected waiting time does not depend on how long you've already waited, that is Prob (X > s+t | X > s) = Prob (X > t) for any s,t greater than 0. - the average waiting time (the expected value of X) is 1/lambda (NOT lambda!). Uniform Distribution on interval [a,b] -Know intuition, pdf, cdf, mean. (Easy!) -Be able to compute variance and standard deviation. Here is the easy way. If X has uniform distribution on [0,1], it's not hard to check the variance E(X^2) - (E(X))^2 is 1/12, and therefore standard deviation square root of 1/12. Since Y = a + (b-a)X has uniform distribution on [a,b], its standard deviation is then (b-a)x(square root of 1/12). Normal Distribution Crucial! The bell shaped curve. Know the pdf. Be able to use the normal distribution table to compute probabilities for any of the normal distributions (not just N(0,1) ). Understand all those normal distributions are essentially like N(0,1), recentered at the mean and rescaled by the standard deviation. Be able to find a percentile of the N(0,1) distribution. For example, the 90th percentile is (to a decimal approximation) equal to 1.645, because if an rv Z has the N(0,1) distribution, the Prob(Z < 1.645) = .90 . Be able to find a percentile of an N(mu,sigma) distribution. For example, its 90th percentile would be the number mu + (1.645)sigma . Again, this looks just like N(0,1) -- recentered at mu, and rescaled by sigma. Continuous random variables with p.d.f. f Know when a pdf f is legal (f is nonnegative and total area under the graph of f is 1). Know when a cdf F is legal (F is continuous, nondecreasing, F(x) goes to 0 at -infinity and goes to 1 at infinity). Know the definition of Prob (a