Econ Stats Cumulative Review

What does it mean for a random variable X to have a cumulative distribution function F?

For any real number x, Pr(X≤x)=F(x).

Suppose that the c.d.f. F of a random variable X is given by F(x)={0 for x<0 ={x^2 for 0≤x≤1 ={1 for x>1. What is the p.d.f. f of X?

f(x)={2x for 0≤x≤1 ={0 otherwise

What should you type in RStudio if you want to find out what your current working directory is?

getwd()

Suppose that a random variable X has the binomial distribution with parameters n and p. What is the expectation of X?

np

What is the value of ∫^1 over 0 xdx?

1/2

If a die is rolled three times, what is the probability that the same number will show up twice in a row (for example, 112 or 633 or 555)?

11/36

What is the value of ∫^1 over 1/2 xdx?

3/8

Suppose we want to run an experiment 3 times where the core experiment is to draw 4 numbers out of the set of integers between 1 and 10 without replacement. How could we do that in RStudio?

replicate(3,sample(1:10,4))

Suppose that one card is to be drawn from a deck of 3 red cards numbered from 1 to 3, and another card is to be drawn from a deck of 3 yellow cards numbered from 4 to 6. What is the sample space S?

{14, 15, 16, 24, 25, 26, 34, 35, 36}

Suppose that one card is to be drawn from a deck of 6 cards that contains 3 red cards numbered from 1 to 3 and 3 yellow cards numbered from 4 to 6. Let A be the event that the card drawn is an even number. Let B be the event that the card drawn is yellow. Let C be the event that the card drawn is a number less than 5. What is the event A∩B∩C?

{4}

A deck contains five cards: the 3, the 7, the Jack, the Queen, and the King. You draw cards from the deck in random order without replacement until a face card (J, Q, or K) is drawn. For example, 73J is one possible sequence of draws (outcome). What is the event that the 7 is not drawn?

{J, Q, K, 3J, 3Q, 3K}

Suppose that the random variable X has the normal distribution with mean 1 and variance 9. What is Pr(X≤4)?

Φ(1)≈0.8413

What is the value of the sum ∑∞ over i=1 α^i, where 0<α<1?

α/1-α

Suppose that in RStudio we want to graph the probability density function of the normal distribution with mean 7 and variance 4. Which of the following commands will do that? I. curve(dnorm(x-7, mean = 0, sd =2), xlim=c(3,11)) II. curve(dnorm(x, mean = 7, sd =2), xlim=c(3,11)) III. curve(0.5*dnorm((x-7)/2, mean = 0, sd =1), xlim=c(3,11))

!, II, and III

A volleyball team will play 12 matches this season. Suppose that the team's probability of winning any given match is 60%, and that winning a given match is independent of winning other matches. Let the random variable 𝑋 equal the team's total number of victories this season. Suppose that the team's coach gets a bonus of $100 for each of the team's victories. Let the random variable 𝑌 equal the total amount of bonus payment to the coach for the season. What is the expectation of 𝑌?

$720

Suppose that a continuous random variable X has the following probability density function: f(x)={4x^3 for 0≤x≤1 ={0 otherwise. What is the median of the distribution of X?

(1/2)^1/4

Suppose that a continuous random variable X has the following probability density function: f(x)={4x^3 for 0≤x≤1 ={0 otherwise. What is the 0.75 quantile of the distribution of X?

(3/4)^1/4

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the mean of X?

-1

Suppose that the random variable X1 has the normal distribution with mean 7 and variance 16, and that the random variable X2 has the normal distribution with mean −1 and variance 4. The covariance of X1 and X2 is −4. What is the correlation ρ(X1,X2) of X1 and X2?

-1/2

A random variable 𝑋 has the uniform distribution on the interval [10,20]. What is the value of 𝑃𝑟(𝑋 =11 )?

0

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the value of Pr(X=1)?

0

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. If the probability that the student has seen both movies is 0.48, then what is the probability that the student has seen Star Wars but not Frozen?

0.02

Suppose that 10,000 tickets are sold in one lottery and 5000 tickets are sold in another lottery. If a person owns 100 tickets in each lottery, what is the probability that she will win at least one first prize?

0.0298

Suppose that 1000 tickets are sold in one lottery and 5000 tickets are sold in another lottery. If a person owns 100 tickets in the first lottery and 400 tickets in the second, what is the probability that he will win at least one first prize?

0.172

A random student from ECO 329 is selected. Let 𝐴 be the event that the student wears glasses, and let 𝐵 be the event that the student is less than 6 feet tall. Suppose that Pr(𝐴)=0.3 and that Pr(𝐵)=0.8. Which of the following is a mathematically possible value of 𝑃𝑟(𝐴∩𝐵)?

0.2

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the value of Pr(0<X<2)?

0.2

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. Which of the following is a mathematically possible value of Pr(A∩B)?

0.4

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. The probability that the student has seen both movies is 0.495. If we know that the student has seen Frozen, then what is the probability that the student has also seen Star Wars?

0.55

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. The probability that the student has seen both movies is 0.495. If we know that the student has seen at least one of the two movies, then what is the probability (to 3 decimal places) that the student has seen Star Wars?

0.552

A random variable 𝑋 has the uniform distribution on the interval [10,20]. What is the value of 𝑃𝑟(𝑋 >13 )?

0.7

A random student from ECO 329 is selected. Let 𝐴 be the event that the student wears glasses, and let 𝐵 be the event that the student is less than 6 feet tall. Suppose that Pr(𝐴)=0.3 and that Pr(𝐵)=0.8. Which of the following is a mathematically possible value of 𝑃𝑟(𝐴∪𝐵)?

0.9

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. If the probability that the student has seen both movies is 0.48, then what is the probability that the student has seen at least one of the movies?

0.92

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. The probability that the student has seen both movies is 0.495. If we know that the student has seen Star Wars, then what is the probability that the student has also seen Frozen?

0.99

A random student from ECO 329 is selected. Let A be the event that the student has seen the movie Frozen, and let B be the event that the student has seen the movie Star Wars. Suppose that Pr(A)=0.9 and that Pr(B)=0.5. Which of the following is a mathematically possible value of Pr(A∪B)?

1

You repeatedly flip a fair coin until either Heads comes up or you have flipped three times, whichever comes first. For example, TTT and TH are two possible sequences of flips (outcomes). Define the random variable 𝑋 as the number of times you flip the coin. What is the expectation of 𝑋?

1.75

Suppose that X1 has the uniform distribution on the interval [0,1], and that X2 has the uniform distribution on the interval [100,101]. What are the variances of X1 and X2, respectively?

1/12 and 1/12

You repeatedly flip a fair coin until either Heads comes up or you have flipped three times, whichever comes first. For example, TTT and TH are two possible sequences of flips (outcomes). Define the random variable 𝑋 as the number of times you flip the coin. What is the variance of 𝑋?

11/16

At a particular bank, 40% of customers are under 25 years old, and 60% are 25 or older. On a given day, 10% of customers under 25 make a withdrawal, and 20% of customers aged 25 or older make a withdrawal. What is the probability that a randomly chosen customer makes a withdrawal?

16%

Suppose that the random variable X has the normal distribution with mean 1 and variance 9. What is Pr(X>7)?

1−Φ(2)≈0.0227

What is the derivative of the function f(x)=2e^x?

2e^x

If a random sample of 32 observations is taken from the normal distribution with mean μ and variance 8, what is the probability that the sample mean will be within 0.75 of μ?

2Φ(1.5)−1≈0.3830

If a coin is flipped three times, what is the probability that the same side will show up twice in a row (for example, HHT or HTT or TTT)?

3/4

At a particular bank, 40% of customers are under 25 years old, and 60% are 25 or older. On a given day, 10% of customers under 25 make a withdrawal, and 20% of customers aged 25 or older make a withdrawal. What is the probability that a customer is under 25 and makes a withdrawal?

4%

Suppose that a continuous random variable X has the following probability density function: f(x)={4x^3 for 0≤x≤1 ={0 otherwise. What is the expectation of X?

4/5

If five dice are rolled, what is the probability that each of the five numbers that appear will be different?

5/54

Suppose that Pete Cutchew has a 70% chance of getting an A on his Physics midterm and a 50% chance of getting an A on his Art History midterm. If those events are positively correlated, which of the following is a mathematically possible value for the probability that Pete gets an A on both midterms?

50%

Suppose that a continuous random variable X1 has expectation 7 and variance 4, and that a discrete random variable X2 has expectation −1 and variance 2. The correlation of X1 and X2 is 0.12. What is the expectation of X1+X2?

6

There are 10 different movies available to watch on a long airplane flight. If first you pick a movie, and then your friend picks a movie, and then your other friend picks a movie, how many possible sequences of three movies are possible without watching the same movie more than once?

720

At a particular bank, 40% of customers are under 25 years old, and 60% are 25 or older. On a given day, 10% of customers under 25 make a withdrawal, and 20% of customers aged 25 or older make a withdrawal. If a customer made a withdrawal, what is the probability that the customer is 25 or older?

75%

Suppose that a random variable X has the binomial distribution with parameters n=20 and p=0.25. How would you use R to find Pr(X=8)?

> .25^8 * .75^12 * factorial(20) / (factorial(12)*factorial(8)) [1] 0.06088669

Suppose that the probability that a randomly selected worker at a particular firm will suffer an on-the-job injury is 0.01. If the firm has 10 workers, and the probabilities of an injury are independent across workers, how would you use R to calculate the probability that at least one worker will suffer an injury?

> 1-(.99)^10 [1] 0.09561792

Suppose that the probability that a randomly selected worker at a particular firm will suffer an on-the-job injury is 0.01. If the firm has 10 workers, and the probabilities of an injury are independent across workers, how would you use R to calculate the probability that exactly one worker will suffer an injury?

> 10*0.01*(0.99)^9 [1] 0.09135172

There are 10 students in a theater class that will perform a play with four roles: Dorothy, the Lion, the Tin Man, and the Scarecrow. The class must also select four students to be members of the school's Student Council. How would you use R to calculate the number of different possible groups of four students?

> factorial(10) / (factorial(6)*factorial(4)) [1] 210

There are 10 students in a theater class that will perform a play with four roles: Dorothy, the Lion, the Tin Man, and the Scarecrow. How would you use R to calculate the number of different ways to cast the play?

> factorial(10) / factorial(6) [1] 5040

Suppose that a random variable X has the binomial distribution with parameters n=10 and p=0.7. How would you use R to find all the values of k such that Pr(X=k)>0.2?

> n <- c(0:10) > n[ .2 < .7^(n) * .3^(10-n) * factorial(10) / (factorial(10-n)*factorial(n))] [1] 6 7 8

Suppose that a random variable 𝑋 has the binomial distribution with parameters 𝑛=10 and 𝑝=0.7. How would you use RStudio to find all the values of 𝑘 such that 𝑃𝑟(𝑋 =𝑘) > 0.2?

> n <- c(0:10) > n[ .2 < .7^(n) * .3^(10-n) * factorial(10) / (factorial(10-n)*factorial(n))]

Suppose that a random variable X has the binomial distribution with parameters n=20 and p=0.25. How would you use R to find Pr(X<9)?

> n <- c(0:8) > pn <- .25^(n) * .75^(20-n) * factorial(20) / (factorial(20-n)*factorial(n)) > sum(pn) [1] 0.9590748

Suppose that 15% of the students in a class studied 10 hours for a midterm exam, 35% studied 5 hours, 40% studied 3 hours, and 10% did not study at all. The probability that a student who studied 10 hours could pass the exam is 100%; the probability for a student who studied 5 hours is 90%; the probability for a student who studied 3 hours is 80%; and the probability for a student who did not study at all is 50%. A student is randomly selected from the class. How would you use R to answer the following question: if she passes the exam, what is the (posterior) probability that she studied 3 hours?

> prior <- c(.15, .35, .4, .1) > pass <- c(1, .9, .8, .5) > z <- prior*pass > z[3] / sum(z) [1] 0.3832335

Suppose that the weights in kilograms of the 7 fish that you caught today are 2.48, 1.74, 8.14, 4.29, 2.37, 3.78, and 4.41. How would you use R to calculate the mean and variance of those weights?

> weight.fish <- c(2.48, 1.74, 8.14, 4.29, 2.37, 3.78, 4.41) > mean(weight.fish) [1] 3.887143 > var(weight.fish) [1] 4.57099

Suppose that the weights in kilograms of the 7 fish that you caught today are 2.48, 1.74, 8.14, 4.29, 2.37, 3.78, and 4.41. How would you use R to calculate the mean of the weights of the fish that are over 3 kg?

> weight.fish <- c(2.48, 1.74, 8.14, 4.29, 2.37, 3.78, 4.41) > mean(weight.fish[weight.fish>3])[1] 5.155

Suppose that the weights in kilograms of the 4 fish that you caught today are 4.28, 7.14, 4.18, and 3.78. How would you use R to calculate the mean of the weights of the fish that are under 5 kg?

> weight.fish <- c(4.28,7.14,4.18,3.78) > mean(weight.fish[weight.fish<5]) [1] 4.08

A random student from ECO 329 is selected. Let 𝐴 be the event that the student wears glasses, and let 𝐵 be the event that the student is less than 6 feet tall. Suppose that Pr(𝐴)=0.3 and that Pr(𝐵)=0.8. How do we write the event that the student is less than 6 feet tall but does not wear glasses?

A^c ∩ B

In R, what is the main difference between a matrix and a data frame?

All the elements of a matrix must be of the same type, but the elements of a data frame may have elements of different type (e.g., some numeric and others text).

Suppose that the state space of an experiment is S={1,2,3,4,5,6,7,8,9,10}. Which of the following is a partition of S?

B1={1,4,7} B2={2,5,8} B3=3,6,9} B4={10}

A volleyball team will play 12 matches this season. Suppose that the team's probability of winning any given match is 60%, and that winning a given match is independent of winning other matches. Let the random variable 𝑋 equal the team's total number of victories this season. What is the distribution of 𝑋?

Binomial with parameters n=𝟏𝟐 and 𝒑=𝟎.𝟔

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the c.d.f. F of X?

F(x)={0 for x<-6 ={(x+6)/10 for -6≤x≤4 ={1 for x>4

What does it mean for a continuous random variable 𝑋 to have a probability density function 𝑓?

For any interval of real numbers [𝒂,𝒃], the probability that 𝑿 takes a value between 𝒂 and 𝒃 is the integral of 𝒇 over the interval.

What does it mean for a continuous random variable X to have a probability density function f?

For any interval of real numbers, the probability that the value of X lies in that interval is found by taking the integral of f over that interval.

Either give an example of two events 𝐴 and 𝐵 that satisfy the condition 𝐴∩𝐵 =𝐴 (a picture is acceptable), or explain why such an example is impossible.

If 𝑨⊂𝑩 (that is, 𝑨 is contained in 𝑩), then 𝑨∩𝑩=𝑨. For example, suppose that we roll a die once, that 𝑨=(𝟏,𝟐) is the event that the number rolled is less than 3, and that 𝑩=(𝟏,𝟐,𝟑) is the event that the number rolled is less than 4. Then 𝑨∩𝑩=(𝟏,𝟐)=𝑨.

Suppose that the random variable has the normal distribution with mean and variance , and that the random variable has the normal distribution with mean and variance . The random variables and are independent. What is the distribution of ?

Normal distribution with mean 6 and variance 6

Suppose that the random variables X1,...,X10 are independent, and that each has the normal distribution with mean 80 and variance 3. What is the distribution of the sample average (X1+⋯X10)/10?

Normal distribution with mean 80 and variance 3/10

A deck contains three cards: the Jack, the Queen, and the King. You randomly draw cards from the deck without replacement until either the King is drawn or you have drawn twice, whichever comes first. For example, QJ and K are two possible sequences of draws (outcomes). What is the sample space?

S={K,QJ, QK, JQ, JK}

A deck contains three cards: the Jack, the Queen, and the King. You randomly draw cards from the deck without replacement until either the King is drawn or you have drawn twice, whichever comes first. For example, QJ and K are two possible sequences of draws (outcomes). What is the event that the Queen is drawn?

S={QJ, QK, JQ}

What does the Law of Large Numbers say?

Suppose that X1,...,Xn form a random sample from a distribution for which the mean is μ and for which the variance is finite. Then when n is very large, the probability that the sample average (X1+⋯Xn)/n is close to μ is close to 1.

What is the outcome if you type > sqrt(64) == 9 | factorial(7) == 5040

TRUE

A volleyball team will play 12 matches this season. Suppose that the team's probability of winning any given match is 60%, and that winning a given match is independent of winning other matches. Let the random variable 𝑋 equal the team's total number of victories this season. How would the answer to the previous question change if the outcomes of the 12 matches were NOT independent?

The answer would not change

A deck contains three cards: the Jack, the Queen, and the King. You randomly draw cards from the deck without replacement until either the King is drawn or you have drawn twice, whichever comes first. For example, QJ and K are two possible sequences of draws (outcomes). What is the probability that the King is drawn?

The support is 𝑺={𝑲,𝑸𝑱,𝑸𝑲,𝑱𝑸,𝑱𝑲}, with probabilities {𝟏/𝟑.𝟏/𝟔,𝟏/𝟔,𝟏/𝟔,𝟏/𝟔}. The event that the King is drawn is {𝑲,𝑸𝑲,𝑱𝑲}, which has probability 1/3 + 1/6 + 1/6 = 2/3

On a particular farm, 1/3 of the animals are pigs and 2/3 are chickens. 30% of the pigs are brown, and 60% of the chickens are brown. Are the events "animal is brown" and "animal is a chicken" independent? Briefly explain why or why not, or explain why we need more information to be able to determine the answer.

They are not independent - the probability of the first event depends on the second event. To put it another way, 𝐏𝐫(𝒃𝒓𝒐𝒘𝒏 & 𝒄𝒉𝒊𝒄𝒌𝒆𝒏) = 𝟎.𝟒 is not equal to 𝐏𝐫(𝒃𝒓𝒐𝒘𝒏) x 𝐏𝐫 (𝒄𝒉𝒊𝒄𝒌𝒆𝒏) = 𝟎.𝟓 x 2/𝟑 =1/𝟑.

What does it mean for a random variable X to have a discrete distribution?

X has a finite or countable set of possible values.

Consider the variable "cty" (miles per gallon in the city) from the data set "mpg" that is included as part of the "ggplot2" package in RStudio. Suppose that we want to create a variable "ctycat" that indicates whether the value of cty is "High" (above 20), "Medium" (between 15 and 20), or "Low" (below 15). How could we do that in RStudio?

ctycat<-ifelse(mpg$cty>20,"High",ifelse(mpg$cty>15,"Medium","Low"))

Suppose that the random variable X has the normal distribution with mean 103 and variance 4. What is the smallest value of c such that Pr(103−c≤X<103+c)≥0.9?

c≈3.29

Suppose that the random variable X has the normal distribution with mean 3 and variance 4. What is the smallest value of c such that Pr(3−c≤X<3+c)≥0.9?

c≈3.29

Suppose that a random variable X has the uniform distribution on the interval [−6,4]. What is the p.d.f. f of X?

f(x)={1/10 for -6≤x≤4 ={0 otherwise

Consider the data frame mpg that is included as part of the ggplot2 package in RStudio. Suppose that you want to create a subset with only the cars from 2008, and you only want to keep the variables, "model," "cyl", and "trans". What command would you use?

mpgHW5 <- subset(mpg,subset=year==2008,select=c("model","cyl", "trans"))

Suppose that you have a data frame named "example" in R. What command would you use to find out what types of elements it contains?

str(example)

Suppose that one card is to be drawn from a deck of 6 cards that contains 3 red cards numbered from 1 to 3 and 3 yellow cards numbered from 4 to 6. Let A be the event that the card drawn is an even number. Let B be the event that the card drawn is yellow. Let C be the event that the card drawn is a number less than 5. What is the event A∪B∪C?

{1,2,3,4,5,6}

You repeatedly flip a fair coin until either Heads comes up or you have flipped three times, whichever comes first. For example, TTT and TH are two possible sequences of flips (outcomes). Define the random variable 𝑋 as the number of times you flip the coin. What is the support of 𝑋?

{1,2,3}

Suppose that one card is to be drawn from a deck of 3 red cards numbered from 1 to 3, and another card is to be drawn from a deck of 3 yellow cards numbered from 4 to 6. Let A be the event that the red card drawn is an odd number. Which is the correct mathematical definition of A ?

{14, 15, 16, 34, 35, 36}

Suppose that one card is to be drawn from a deck of 3 red cards numbered from 1 to 3, and another card is to be drawn from a deck of 3 yellow cards numbered from 4 to 6. Let A be the event that the red card drawn is an odd number. Which is the correct mathematical definition of A^c?

{24, 25, 26}

Suppose that one card is to be drawn from a deck of 6 cards that contains 3 red cards numbered from 1 to 3 and 3 yellow cards numbered from 4 to 6. Let A be the event that the card drawn is an even number. Let B be the event that the card drawn is yellow. Let C be the event that the card drawn is a number less than 5. What is the event B∪C^c?

{4,5,6}

A deck contains five cards: the 3, the 7, the Jack, the Queen, and the King. You draw cards from the deck in random order without replacement until a face card (J, Q, or K) is drawn. For example, 73J is one possible sequence of draws (outcome). What is the sample space S ?

{J, Q, K, 3J, 3Q, 3K, 7J, 7Q, 7K, 37J, 37Q, 37K, 73J, 73Q, 73K}

A deck contains five cards: the 3, the 7, the Jack, the Queen, and the King. You draw cards from the deck in random order without replacement until a face card (J, Q, or K) is drawn. For example, 73J is one possible sequence of draws (outcome). What is the event that the King is drawn?

{K, 3K, 7K, 37K, 73K}

Suppose that the random variable X has the normal distribution with mean 1 and variance 9. What is Pr(−2≤X<2.5)?

Φ(1/2)−Φ(−1)≈0.5328