Midterm 1 CS 361

What is the probability of getting 2 hearts if one deals 7 different cards out of a fairly shuffled deck of standard poker?

( 13C2 * 39C5 ) / 52C7

Which of the following is true about the correlation coefficient of the data set {(x,y)} = {(2,1), (4,2.5), (6,1.2)}?

(a) Their correlation coefficient is positive and if both x and y are scaled with a factor of -2, their correlation coefficient remains positive.

Which of the following numbers can be a probability value?

(d) 0.5

Suppose a pet store has two kinds of goldfish, one is orange colored and the other is blue colored. The assistant shows me 10 blue ones and 30 orange ones. I would like to pick 8 randomly with replacement. If X is the number of blue goldfish that are picked and Y is the number of orange goldfish that are picked. What is E[X−Y]?

-4

Suppose random variables X1,X2,X3, are from the same distribution for which the expected value is 4 the value of E[2X1-3X2+X3-4] is

-4

A random variable X has the probability distribution such that P(X=x) = {P 1/3 x=-2, 1/3 x=-1, 1/3 x=0, 0 otherwise. What is E[6|x|+1]?

-5 because 6[1/3 * (|-2|) + 1/3 (|-1)] + 1

What is var({mean({xi+1})})?

0

Suppose a student has 50% chance of contracting the COVID virus if going out without wearing mask, and the chance of contracting COVID is 5% if the student is wearing mask. If the student wears mask 70% of the times while going out, what's the probability the student contracts COVID? Choose the closest value among the following:

0.18

A biased four-sided die is rolled once. Random variable X is defined to be the down-face value. The probability function of X is such that P(X=x) = {x/10 x=1,2,3,4, 0 otherwise} What is P(X<=3)?

0.6

A roulette wheel has 36 slots numbered 1-36. Of these slots, the odd numbers are red and the even numbers are black. In addition, there are two slots numbered zero, which are green. The croupier spins the wheel, and throws a ball onto the surface; the ball bounces around and ends up in a slot (which is chosen fairly and at random). What is closest to the probability of the ball falling into an even numbered slot (consider 0 as even number)?

10/19

A number lock has 4 dials such that each can be turned to any one of the 10 digits in [0,9], how many choices are there to set the lock? 10!/6! 10C4 4^10 10^4

10^4

A big bowl contains 3 red chips and 6 white chips. A chip is drawn at random. If it is red, that chip along with an additional red chip will be put back in the bowl. If it is white, the chip is simply returned to the bowl. Next a second chip is drawn. What is the probability that the 2nd chip is red?

16/45

We toss identical fair coins A&B independently 2 times and 5 times respectively. For each head, we earn $1. Suppose X is the earning from A and Y to be the earning from B. . What is E[XY]?

2.5

Suppose the college has 15 identical gifts for 21 alumini. What is the size of the sample space for the possible arrangement? Suppose each person can only get one gift.

21C15

A random variable X has the probability distribution such that P(X=x) = { 1/3 x=-2, 1/3 x=-1, 1/3 x=2, 0 otherwise. What is var[3X]

26 because 9[varx]=9[2^2*1/3 + (-1)^2 * 1/3 + (-2)^2 * 1/3

Among 5-bit binary numbers, how many have at least 4 consecutive "1"s?

3

A random variable X has the probabiliy distribution such that {1/3 x=-2, 1/3 x=-1, 1/3 x=0, 0 otherwise. What is E[3X^2]?

5 because 3(1/3 * (-2)^2 + 1/3 (-1)^2 + 1/3 (0))

Professor Hongye has a list of 55 different songs. In the coming week, shje wants to randomly select 3 songs to play on Wed and then select 3 songs that haven't been selected on Wed to play on Fri. How many ways are there for her arrangements? It doesn't matter how the 3 songs will be played on a day.

55!/3!*3!*49!

How many five-letter code words are possible using the letters in "DeMorgan", if the letters may NOT be repeated? 8^5 5^8 8!/3! 8!/3!5!

8!/3!

A big bowl contains 3 red chips and 6 white chips. A chip is drawn at random. If it is red, that chip along with an additional red chip will be put back in the bowl. If it is white, the chip is simply returned to the bowl. Next a second chip is drawn. If the 2nd chip is white, what is the probability that the first is red?

9/29

On a certain day, an airline sells a kind of special ticket that gives the customer a free meal. Suppose there is one such ticket in every 100 tickets. The tickets are shuffled randomly before being sold. What is the expected number of usual tickets are sold before such a special ticket is obtained?

99

Which of the following are true? Events on a probability set can be treated as sets and as such normal set operations can be used on events P((A U B)^c) = P(A^c | B^c)P(B^c)

Both

Which of the following are true? If X is a discrete random variable, |X| is also a discrete random variable. discrete random variable, the sum of the probabilities of all possible valuies X takes is always 1

Both

Which of the following are true? The weak law of large numbers justifies using the histogram to represent the distribution of a random variable. The weak law of large numbers justifies using simulation to estimate the expected values of random variables

Both

Statement 1: Anna randomly picks a day to bike in 2020, the event she bikes on any Sunday and the event she bikes on any 15th of a month are disjoint Statement 2: Roll a 4-sided die twice the event "the first die comes up 2" and "the first die comes up 4" are independent.

Both are incorrect

Which of the following are true? A random variable is a function of all outcomes in a sample space. If X is a random variable, Y=sin(X), then E[Y]=sin(x)P(X=x)

Both of them

Which of the following two statements are correct? (1) One can do set operations on events as they can be represented as sets of outcome. (2) P((A N B) ^C) = P(A^C) + P(B^C) - P(A^C N B^C)

Both statements

What are the values of E[E^2[X]] and var[E^2[x]] respectively?

E^2[X], 0

True or False: if two one-dimensional data sets have perfect linear relationship, their correlation coefficient should be 1.

False

The following are true? The event a student is sitting in a classroom and the student is on the Canvas site for chat are disjoint. T A and B have positive probabilities and P(A|B) = P(B|A) then P(A) is the same as P(B)

None

Which of the following are true? The event a student is in class CS361 and the event the student is taking it remotely are disjoint. When two random events are independent and they have non-zero probability, they are disjoint.

None of them

Suppose I randomly draw one card from a standard card deck, which of the following are true? The event {that the drawn card is King} and the event {the drawn card is Queen} are disjoint. The above two events are independent.

Only the first

Suppose random variable X and Y are negatively correlated, which of the following are true? 1) The covariance of X and Y is negative 2) var[X+Y] is larger than var[X-Y]

Only the first

Which of the following are true about the statistics of a data set? If we have a set of 3D data points, no matter how we translate any of the axes, the projected points on X axis as a 1D data set has the same standard deviation. Since the mean is more sensitive to outliers than the median, the mean of a data set is useless.

Only the first

Which of the following are true about the statistics of a data set? The mean is more sensitive to outliers than the median. Var({2xi}) = 2var({xi})

Only the first

Which of the following are true? The expected value of a Bernoulli random variable with p=0.8 is 0.8. When two discrete random variables X and Y have identical independent probability distribution, then 2X+Y is equivalent to 3X regarding their probability function.

Only the first

Which of the following are true? The square of a Bernoulli random variable is a random variable as well. When two discrete random variables X and Y have identical independent probability distribution, then X+Y is equivalent to 2X regarding their probability function.

Only the first

Which of the following are true? P(A|B)=P(B|A)P(A) where P(A) and P(B) are non-zero P(A|B)>=P(A N B), where P(B) is non-zero

Only the second

Which of the following are true? The Chebychev inequality can be used to explain why 90% or more data points of a standardized data set lie within [-1,1] The weak law of large numbers assumes the large numbers of samples or random variables involved are iid samples or random variables.

Only the second

Statement 1: Pairwise independence between a set of events (more than 2 events) is equivalent to mutual independence Statement 2: Maximum of a discrete CDF function is 1

The first is incorrect, the 2nd is correct.

A data set {xi} = {-5,-1,0,0,0,1,3,5}, if a data item {0.5} is added to it, which of the following is true?

The mean increases, the median doesn't change

True or False: If we have a data set of two companies' stock prices in Sept. 2020, and we scale both of them with -5, the correlation between the stocks of the two companies won't change

True

True or false: If event A and B^c are mutually exclusive, P(A U B^c) = P(A)-P(B)+1

True

Estimate as close as possible, 70% of a standardized data set is within: [-5,5] [-2,2] [-10,10] [-1.5,1.5]

[-2,2]

Estimate as close as possible, 80 percent of a standardized data set is within: [-5,5] [-10,10] [-2.5,2.5] [-4,4]

[-2.5,2.5]

Which of the following are true about the statistics of a data set? Correlation coefficient of two data sets can never be 2 Interquartile range of a data set is a scale parameter

both

Which of the following is always true about correlation and correlation coefficient?

correlation coefficient of 2 random features lies in [-1, 1].

The probability of triplets in human births is approximately 0.001. If we use the Poisson model, what is the probability that there will be exactly one set of triplets among 700 births in a large hospital? Consider one set of triplets as one birth.

e^-0.7 *0.7

Which of the following is always true for Inter Quantile Range (IQR)?

iqr{xi-2}=iqr{xi}

Which of the following is always true? std({2xi})=2std({xi}) std({2|xi|})=2std({|xi|}) std({xi+2})=std({xi})+2 std({xi^2})=std({xi})^2

std({2xi})=2std({xi})

Is median a location parameter?

true

Which is correct about the range of the values of the random variable D=X-Y and the probability of P(D=0)? X and Y are two independent Bernoulii random variables with probability p= 0.5.

{-1,0,1} & P(D=0) = 0.5

Which is correct about the range of the values of the random Z=X*Y and the probability of P(Z=0) while X and Y are two independent Bernoulii random variables with probability p=0.5

{0,1} & P(Z=0)=0.75