advertisement

- Modified Jeopardy Name that Continuous Distribution Rules for Expectations Convergence Related Name that Discrete Distribution Xforms of All Types 100 100 100 100 100 200 200 200 200 200 300 300 300 300 300 400 400 400 400 400 500 500 500 500 500 What distribution might be appropriate for jointly modeling test scores for two exams where each exam’s marginal distribution was symmetric? What distribution would be appropriate to model the proportion of voters who favor a candidate? What distribution would be appropriate for modeling response times for a machine where you expect most times to be short and few to be large, where time is not measured in discrete units? What distribution might be appropriate for modeling height of giraffes in zoos? (Height is a common example where this distribution occurs in practice). This is the only continuous distribution with the memoryless property and it also can help model waiting time between Poisson events. If X is independent of Y, Var(X)=2, and Var(Y)=8, what is Var(6X+2Y)? If Y has E(Y)=4 and SD(Y)=3, and X=4Y+9, what are E(X) and SD(X)? If X and Y are jointly distributed RVs, Cov(X,Y)=0, E(X)=6, E(XY)=12, what is E(Y)? Does Cov(X,Y)=0 imply X is independent of Y? If Y has E(Y)=2 and SD(Y)=1 and X=Y^2-6Y+3, what is E(X)? Is there enough information given to get SD(X)? X and Y are jointly distributed RVs with correlation .5, V(X)=10, and V(Y)=6. What is Var(2X-Y)? The CLT is an example of this type of convergence result. If we have a random sample of n observations, where E(X) and V(X) exist, then X-bar converges in quadratic mean to this value. If X is Gamma(2,3), and we have a random sample of 30 variables that behave like X, what does X-bar converge in probability to? How do you justify this? If X is Poisson(6), then Y=X-6/sqrt(6) converges in distribution to this distribution. If X is from any unknown distribution with E(X) and V(X) finite, what can you say about the distribution of X-bar for a random sample of 60 variables from X’s distribution? What distribution would be appropriate to model the number of trains arriving at an Amtrak station in an hour if in the past year, there have been 3 trains hour arriving, on average, each hour? What distribution would be appropriate to model the number of times a person takes a shot in basketball before making a total of five baskets? What distribution would be appropriate for modelling the size of a population of wild animals based on a capture-recapture method? What distribution would be appropriate for modeling t-shirt sales for a store where there are 4 shirt sizes, and they plan to order 900 shirts? What distribution would be appropriate to model the number of successful police stings out of a fixed number of planned stings in a given month assuming the stings are independent? If Y_1,Y_2,Y_3, and Y_4 are all Geometric(.7), what distribution does the sum of the Y’s have? (How do you know?) If Y is Uniform(6,10), what are appropriate methods to find the pdf of X=Y^2+6? If X and Y are independent, each Exp(2), and Z=X+Y, and W=X, what methods will give you the joint density for Z and W? If X and Y are independent Gamma RVs, what methods exist to find the pdf of W=2X+3Y? The neat matrix required to do the method of 2-D xforms. Standardization Fun Pesky Integrals Probability Basics Anything Conditional Random 200 200 200 200 200 400 400 400 400 400 600 600 600 600 600 800 800 800 800 800 1000 1000 1000 1000 1000 The value you would compute if you wanted to find P(X>16) when X was normally distribution, mean 12, SD 4. If X is Binomial (1000,.4), this is the method you could use to approximate P(X>425). The extra “boost” to any continuous approximation of a discrete distribution, designed to improve the approximation. If X has mean 20 and standard deviation 2, and we have a random sample of 36 observations from X’s distribution, this is how you would find P(X-bar<19). What are good values for n and p for the Normal approximation to the Binomial, and good values for lambda for the similar approximation to the Poisson? For this category, let X and Y denote random variables with joint pdf given by 2 x 2e f ( x, y ) x Where 0<y<x< infinity You may refer to the joint just as f(x,y) but bear the bounds in mind for this category. Setup an integral (or integrals) to find P(Y>5, X>4) Setup an integral (or integrals) to find P(Y<3, X<6) Setup an integral (or integrals) to find P(X>2Y) Setup an integral (or integrals) to find E(X) Setup an integral (or integrals) to find E(XY) These are the three axioms of probability. If you are trying to carry 10 total objects and you decide to give 2 randomly to a friend to carry, this is the number of total object arrangements your friend might end up with. If license plates had to have the following format (L=letter, N=number): LLNNLL, and repeated letters and digits were not allowed, how many possible license plates could be made? If you were asked for P(B|A) and all you had was marginal info about B, B complement, A given B, and A given B complement, this is how you would find the desired probability. If you have just taken a test for a rare disease and received a positive test result, this is the value you should focus on to determine the probability you actually have the disease. If X and Y are independent, then the pdf of X given Y is equivalent to this pdf. If X given Y is Normal(Y, 2), and Y is Uniform(0,4), then E(X) is equal to this value. If X given Y is Normal(Y, 2), and Y is Uniform(0,4), then V(X) could be found by computing this expression (or using this technique) Three prisoners are up for parole, but only 2 will get it. Prisoner C asks a guard for the name of one of the prisoners who is going to be paroled. The guard says “Prisoner A”. Prisoner C is very unhappy as his chances of parole are now ½ instead of 2/3 because either he or prisoner B will get released in addition to A. What if anything is wrong with Prisoner C’s reasoning? The conditioning method of xforms relies on the fact that a joint pdf can be rewritten as this product. A Markov chain with 3 absorbing sets in addition to the entire sample space is called this. If X is Uniform(0,100), this theorem could be applied to find an interval in which 75% of the distribution of X lies. The only discrete distribution that is memoryless Bivariate normal random variables have marginal distributions which have this distribution. (Note you don’t need these formulas, but knowing the distribution is a neat fact.) Martingales, Markov Chains, and Poisson processes are all examples of this type of process. Final Exam is Saturday, December 18th, 2-5 in Merrill 03 You can bring a two-sided page of notes and calculator, plus pen/pencils. Exam Week Office Hours: ◦ Wednesday 3-5 ◦ Thursday 1-4 ◦ Friday – 1-4 Good luck studying! Remember Math dept. end of semester party is today from 3:30 – 4:30 in SM 208! You should be able to access the evaluations online from the course website. I cannot see what you enter there, and will only see it after grades are entered and with names removed. DO NOT HIT ENTER as you type through it as that submits the form. You must use TAB or the mouse to move between fields. (I didn’t set this up.)