- Modified Jeopardy

Name that Continuous Distribution Rules for Expectations Convergence Related Name that Discrete Distribution Xforms of All Types

 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

 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  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.)