- Modified Jeopardy. Name that Continuous Distribution Rules for Expectations Convergence Related Name that Discrete Distribution Xforms of All Types.

Probability Jeopardy Review

- Modified Jeopardy

Name that

Continuous

Distribution

Rules for Expectations

Convergence Related

Name that Discrete

Distribution

Xforms of All Types

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200 200 200 200 200

300 300 300 300 300

400 400 400 400 400

500 500 500 500 500

Categories

What distribution might be appropriate for jointly modeling test scores for two exams where each exam’s marginal distribution was symmetric?

Name that Continuous Distribution (100)

What distribution would be appropriate to model the proportion of voters who favor a candidate?

Name that Continuous Distribution (200)

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?

Name that Continuous Distribution (300)

What distribution might be appropriate for modeling height of giraffes in zoos?

(Height is a common example where this distribution occurs in practice).

Name that Continuous Distribution (400)

This is the only continuous distribution with the memoryless property and it also can help model waiting time between Poisson events.

Name that Continuous Distribution (500)

If X is independent of Y, Var(X)=2, and Var(Y)=8, what is Var(6X+2Y)?

Rules for Expectations (100)

If Y has E(Y)=4 and SD(Y)=3, and X=4Y+9, what are E(X) and SD(X)?

Rules for Expectations (200)

If X and Y are jointly distributed RVs, Cov(X,Y)=0, E(X)=6, E(XY)=12, what is E(Y)?

Rules for Expectations (300) *

Does Cov(X,Y)=0 imply X is independent of Y?

Rules for Expectations (300) D

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)?

Rules for Expectations (400)

X and Y are jointly distributed RVs with correlation .5, V(X)=10, and V(Y)=6. What is Var(2X-Y)?

Rules for Expectations (500)

The CLT is an example of this type of convergence result.

Convergence Related (100)

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.

Convergence Related (200)

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?

Convergence Related (300)

If X is Poisson(6), then Y=X-6/sqrt(6) converges in distribution to this distribution.

Convergence Related (400)

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?

Convergence Related (500)

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?

Name that Discrete Distribution (100)

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?

Name that Discrete Distribution (200)

What distribution would be appropriate for modelling the size of a population of wild animals based on a capture-recapture method?

Name that Discrete Distribution (300)

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?

Name that Discrete Distribution (400)

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?

Name that Discrete Distribution (500)

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?)

Xforms of All Types (100)

If Y is Uniform(6,10), what are appropriate methods to find the pdf of X=Y^2+6?

Xforms of All Types (200)

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?

Xforms of All Types (300)

If X and Y are independent Gamma RVs, what methods exist to find the pdf of W=2X+3Y?

Xforms of All Types (400)

The neat matrix required to do the method of 2-D xforms.

Xforms of All Types (500)

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

Double Categories

The value you would compute if you wanted to find P(X>16) when X was normally distribution, mean 12, SD 4.

Standardization Fun (200)

If X is Binomial (1000,.4), this is the method you could use to approximate P(X>425).

Standardization Fun (400)

The extra “boost” to any continuous approximation of a discrete distribution, designed to improve the approximation.

Standardization Fun (600)

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).

Standardization Fun (200)

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?

Standardization Fun (1000)

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.

Pesky Integrals (Setup)

x

eyxf

x22),(

Setup an integral (or integrals) to find P(Y>5, X>4)

Pesky Integrals (200)

Setup an integral (or integrals) to find P(Y<3, X<6)

Pesky Integrals (400)

Setup an integral (or integrals) to find P(X>2Y)

Pesky Integrals (600)

Setup an integral (or integrals) to find E(X)

Pesky Integrals (800)

Setup an integral (or integrals) to find E(XY)

Pesky Integrals (1000)

These are the three axioms of probability.

Probability Basics (200)

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.

Probability Basics (400)

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?

Probability Basics (600)

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.

Probability Basics (800)

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.

Probability Basics (1000)

If X and Y are independent, then the pdf of X given Y is equivalent to this pdf.

Anything Conditional (200)

If X given Y is Normal(Y, 2), and Y is Uniform(0,4), then E(X) is equal to this value.

Anything Conditional (400)

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)

Anything Conditional (600)

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?

Anything Conditional (800)

The conditioning method of xforms relies on the fact that a joint pdf can be rewritten as this product.

Anything Conditional (1000)

A Markov chain with 3 absorbing sets in addition to the entire sample space is called this.

Random (200)

If X is Uniform(0,100), this theorem could be applied to find an interval in which 75% of the distribution of X lies.

Random (400)

The only discrete distribution that is memoryless

Random (600)

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.)

Random (800)

Martingales, Markov Chains, and Poisson processes are all examples of this type of process.

Random (1000)

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!

Reminder:

Remember Math dept. end of semester party is today from 3:30 – 4:30 in SM 208!

Thanks for a Great Semester!

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