Question

Multiple Choice

Select the best answer from the available choices for each question.

1.   Three people are acting in a play. Character A has 50% of the lines, Character B has 40%, and Character C has 10%. Each person has a different probability of making a mistake on any line, independently: 0.03 for A, 0.07 for B, and 0.20 for C. Given that a mistake occurred, what is the probability that it was C’s line?

  
•   0.02
  
•   0.063
  
•   0.317
  
•   0.444
  
•   None of the above

2.   Which matches the conditions for using a Binomial approximation to the Hypergeometric?

  
•   N must be large and n must be small relative to N
  
•   The sampling must be done with replacement
  
•   n must be large and r/N must be small
  
•   Two or more of the above
  
•   None of the above

3.   Suppose the amount of data you use on your phone (in units of 100 MB) has a Poisson distribution with mean 8 per month. You pay $15 per month plus $3 per 100 MB of data. Find the standard deviation of a random month's phone bill.

•   6.245

•   8.485

•   9.327
  
•   72
  
•   None of the above

4.   The amount of time a user plays Animal Crossing has an Exponential distribution with mean 30 minutes. Given that the user has been playing for 1 hour, what is the probability they will still be playing half an hour later?

•   0.012

•   0.135

•   0.368

•   0.632

•   None of the above

5.   Which of the following 4 statements is FALSE?

  
•   If X~Y (i.e. X and Y have the same distribution), then they have a correlation of 1
  
•   If X and Y are independent, they have a correlation of 0

•   If X and Y have a correlation of 0, and Y and Z have a correlation of 1, then X and Z have correlation of 0

•   Correlation tells us the strength and direction of the linear relationship between two variables, whereas covariance only tells us the direction

•   Two or more of the above are false

6.   Which of the following 4 statements is TRUE?

•   If X and Y have a negative correlation, they are independent

•   If X and Y have a correlation of 0, they are independent

•   If X and Y have a positive correlation, and Y and Z have a positive correlation, then X and Z have positive correlation

•   If X = -0.5Y, then the correlation of X and Y is -1

•   Two or more of the above are true

7.   Which of the following 4 statements is FALSE?

  
•   The Normal distribution has mean, median, and mode all equal to μ   

•   A Normal random variable with a larger variance has a smaller maximum height of its pdf

•   Approximately 95% of the area under the Standard Normal pdf is between -1 and 1

•   The cdf F(x) of a Normal random variable is strictly increasing for all Real values of x

•   Two or more of the above are false

8.   The return on stock X has mean 5 and standard deviation 3. The return on stock Y has mean 8 and standard deviation 10. The standard deviation of an equal portfolio of the two stocks (that is, 0.5X + 0.5Y) is 24.25. What is the correlation between X and Y?

•   -0.2
  
•   0

•   0.2

•   0.4
  
•   None of the above

9.     Suppose the length in cm of a male soccer player's foot follows a Normal distribution with mean 30 and variance 25. Suppose the length in cm of a female soccer player's foot follows a Normal distribution with mean 26 and variance 16. A male and female soccer player are selected at random. What is the probability the female player has a longer foot than the male player?

•   0.09

•   0.27

•   0.33

•   0.73

•   0.91

10.   Independently of one another, a randomly surveyed individual can be a non-smoker, a light smoker, or a heavy smoker with probabilities 60%, 30% and 10%, respectively. Suppose we randomly survey a sample of 100 people. Given that 37 of them are light smokers, what is the expected number of heavy smokers?

•   6.3

•   9

•   27

•   54

•   None of the above

11.   The Princess Theatre has 280 seats. 200 tickets are sold for the new popular movie: The Attack of the Goose. Despite every attendee having an assigned seat, they all decide to sit in seats at random. Find the variance of the number of people sitting in their assigned seat. (Hint: use indicator variables)

•   0.712

•   0.714

•   5.246

•   11.704

•   None of the above

Answer 1

1. P(A)=P(Character A)=0.50, P(B)=P(Character B)=0.40, P(C)=P(Character C)=0.10

M=event of mistake then P(M|A)=0.03, P(M|B)=0.07, P(M|C)=0.20

P(M)=P(M|A)P(A)+P(M|B)P(B)+P(M|C)P(C)=0.03*0.50+0.07*0.40+0.20*0.10=0.063

P(C|M)=[P(M|C)P(C)]/P(M)=(0.20*0.10)/0.063=0.317

Option: 0.317

2. N must be large and n must be small relative to N

3. Y=month's phone bill=15+3X

where X follows Poisson distribution with mean=8

We know, for Poisson distribution, mean=Variance=8

Option: 8.485.

4.

Option: 0.368.

5. Option: If X~Y (i.e. X and Y have the same distribution), then they have a correlation of 1

(If X and Y are independent then they have a correlation of 0).

6. Option: If X and Y have a positive correlation, and Y and Z have a positive correlation, then X and Z have positive correlation.

7. Option: Approximately 95% of the area under the Standard Normal pdf is between -1 and 1

(correct range is [-2,2]).