HW1 SP2024

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University of Pennsylvania *

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102

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Statistics

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Feb 20, 2024

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HW 1 Revision of Probability 1. X , Y and Z are independent random variables with E ( X ) = 2, E ( Y ) = 0, E ( Z ) = - 1 and SD ( X ) = 1, SD ( Y ) = 2 and SD ( Z ) = 3. (a) E ( 3 - X + 2 Y + 4 Z ) = (b) V ( 4 - X + 2 Y - 3 Z ) = (c) SD ( 4 - X + 2 Y - 3 Z ) = (d) E ( X * Y * Z ) = (e) Calculate the SD ( X * Y ) 2. The current age (in years) of 300 clerical employees at an insurance claims processing center is normally distributed with mean 35 and SD 5. (a) Are most of the employees at this center are older than 25? Why or why not? (b) Is it true that if none of these employees leave the firm and no new hires are made, then the distribution a year from now will be normal with mean 36 and SD 6? Why or why not? (c) Three individuals are drawn at random from this population. What are the chances that the first person is younger than the sample average of the second and third individuals? 3. The number of packages handled by a freight carrier daily is normally distributed. On average, 1,200 packages are shipped daily, with SD 150. Assume that the package counts are independent from one day to the next. (a) What is the distribution of the total number of packages shipped over five days? (b) What are the chances that the total number of packages shipped over five days is more than 6500? (c) What is the distribution of the sample mean of the number of packages shipped over ten days? (d) What are the chances that the sample mean of the number of packages shipped over ten days is more than 1150? (e) What is the distribution of the difference between the numbers of packages shipped on any two consecutive days? (f) The difference between the numbers of packages shipped today and the number of packages shipped tomorrow is equal to zero. Would you agree with this statement? (g) If each shipped package earns the carrier $10, then what is the distribution of the amount earned per day? (h) What are the chances that more packages are handled tomorrow than today? 4. A taxi driver charges his passengers a $5 flat fee plus $3 per mile. Suppose that the distribution of trip distances of his passengers is approximately normal with a mean of 10 miles and a standard deviation of 6 miles. (a) What is the distribution of the fares? (b) What is the probability that the fare will exceed $45? (c) Suppose we collect information from 10 passengers (who used this taxi service) on how much money they had spent on total. If Y = total amount spent by these 10 individuals, then i. Y = X 1 + ... + X 10 , where X i s are the amount spent by the ith customer and they are i.i.d ii. Y = 10 X , where X = amount spent by an individual customer iii. Y = X 10 , where X = amount spent by an individual customer iv. Y = X + X + X + X + X + X + X + X + X + X , where X = amount spent by an individual customer v. None of the above (d) What are the expected value and variance of Y? (e) What would be the distribution of Y ? (f) Let M be the sample mean of the information collected from these 10 individuals. What is the distribution of M ? (g) What are the chances that out of the 10 individuals at least one of them spent more than $50. 5. If X is a continuous random variable, then which one of the following options are true, P ( b < X < a ) = (a) P ( b < X a ) 1
(b) P ( b X < a ) (c) P ( b X a ) (d) All of the above (e) None of the above 6. The term random sample implies that... (a) All the observations in the sample are drawn from the same population (b) All the observations in the sample are drawn independently of one another (c) All the elements in the population had an equal chance of being included in the sample drawn (d) All of the first three options are correct (e) None of the first three options are correct 7. If X i s for i = 1 ,..., n are i.i.d ( μ , σ ) , then... (a) E ( n i = 1 X i ) = n μ (b) SD ( n i = 1 X i ) = n σ (c) E ( ¯ X ) = μ (d) SD ( ¯ X ) = σ / n (e) All the four options above are correct (f) None of the first four options are correct 8. Which of the following are true (select all that apply): (a) V ( X ) = V ( - X ) (b) V ( X ) = - V ( X ) (c) E ( X ) = E ( - X ) (d) E ( X ) = - E ( X ) (e) V ( 4 X + 1 ) = 4 V ( X )+ 1 (f) V ( 4 X + 1 ) = 16 V ( X ) (g) V ( 4 X + 1 ) = 16 V ( X )+ 1 (h) E ( 3 X + 7 ) = 3 E ( X )+ 7 (i) E ( 3 X + 7 ) = 9 E ( X )+ 7 (j) E ( 3 X + 7 ) = 9 E ( X ) (k) V ( 3 X - 2 Y + 4 ) = 3 V ( X ) - 29 V ( Y )+ 4 (l) V ( 3 X - 2 Y + 4 ) = 9 V ( X )+ 4 V ( Y )+ 4 (m) V ( 3 X - 2 Y + 4 ) = 9 V ( X ) - 4 V ( Y )+ 4 (n) V ( 3 X - 2 Y + 4 ) = 9 V ( X )+ 4 V ( Y ) (o) V ( 3 X - 2 Y + 4 ) = 3 V ( X ) - 29 V ( Y )+ 4 (p) V ( 3 X - 2 Y + 4 ) = 9 V ( X )+ 4 V ( Y )+ 4 (q) V ( 3 X - 2 Y + 4 ) = 9 V ( X ) - 4 V ( Y )+ 4 Hint: What are the general results? 9. E ( ¯ X ) = μ and SD ( ¯ X ) = σ n tells us that... I. as n V ( ¯ X ) 0, implying that the variation in the observed values of the random variable ¯ X decreases as the sample size increases. II. the distribution of ¯ X has an expected value of μ ONLY when n , the sample size, is sufficiently large (a) (I) is right and (II) is not (b) (II) is right and (I) is not (c) Both (I) and (II) are right (d) Neither (I) nor (II) are right 10. If X i are i.i.d N ( μ , σ ) then the distribution of ¯ X - μ s / n follows a (a) N ( 0 , 1 ) 2
(b) N ( μ , σ ) (c) T distribution with 2 degrees of freedom (d) T distribution with n - 1 degrees of freedom (e) None of the above 11. The key difference between a continuous random variable and a discrete variable is: (a) A continuous random variable can take decimal values and a discrete random variable always takes integer values. (b) A continuous random variable takes infinitely many values and a discrete random variable always takes finitely many values. (c) A continuous random variable takes values over an interval and a discrete random variable always takes values over a discrete set. (d) A continuous random variables takes both positive and negative values and a discrete random variable always takes only positive value. (e) None of the above 12. Which one of the following statements is true (a) If P ( B ) = 0 then the event B must be an impossible event. (b) If C is an impossible event then P ( C ) = 0. (c) For a continuous r.v. X , P ( X = a ) > 0 for certain values observed values by the random variable X . (d) If X is continuous random variable then P ( X a ) > P ( X < a ) (e) None of the above 13. Which one of the following statements is true: (a) The probability density function evaluated at a point " a " gives us P ( X = a ) , where X is the continuous r.v. associated with the density function. (b) The values of a probability density function must always lie between 0 and 1. (c) The probability density function is not a probability but it helps us to calculate probabilities of continuous random variables. (d) Probability density functions can take negative values. (e) None of the above 14. Which one of the following statements is false: (a) For a continuous random variable, X , we have P ( X a ) = P ( X < a ) (b) For a continuous random variable, X , we have P ( b X a ) = P ( b < X a ) = P ( b X < a ) = P ( b < X < a ) (c) A continuous variable observes values over an interval. (d) If X is a continuous random variable then since P ( X = a ) = 0 therefore the event { X = a } is an impossible event. (An impossible event is a set which is not a non-empty subset of the sample space and is denoted by the null set {} ) (e) None of the above. 15. From a population a random sample of diamonds are drawn and certain characteristics are noted. Open the data set diamond in JMP (a) What is the population under considertion? (You have to use your imagination here) (b) What is the size of the sample drawn? (c) What are the individual units in this population? (d) How many different variables are recorded for each of these units? (e) Which ones are Discrete, Cont, Nominal and Ordinal? (f) Using JMP provide a graphical summary of the distribution of the variable price (g) What is the estimate of the population mean of the variable price ? (h) Find the estmate of standard error of the above estimator? (i) State the formula used to calculate the answer in the above question. (j) Based on the formula what would be the value of the standard error as the sample size goes to infinity? 16. Based on a random sample it was found that the estimate of the standard error of the sample mean is equal to 1, we are also told that the estimate of the population standard deviation is equal to 5, what was the size of the random sample? 3
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