IE 3301-001 Engineering Probability Test #1 Practice Questions 1. Given the following data 4 2 1 1 5 3 0 1 0 2 calculate: (a) the sample mean (b) the sample median (c) the sample range 2. For a data sample of 17 observations, we have 34171=∑=iix and 4681712=∑=iix . Calculate: (a) the sample mean (b) the sample standard deviation (c) the coefficient of variation 3. The ages in years of a sample of 25 people in a city park resulted in the following box and whisker plot: Explain and interpret the plot. Does the distribution appear to symmetric, right-skewed, or left-skewed? 4. Consider the letters in the word “MISSISSIPPI.” Write out the correct formulations (do not calculate) for the following: (a) What is the probability of choosing an “I” letter? (b) Suppose you select 2 letters. Given that the first letter is an “S” letter, what is the probability that the second letter is also an “S” letter? (c) How many ways are there to choose 2 “I” letters? How many ways are there to choose 2 “I” letters, one “M” letter, and one “S” letter? (d) How many ways are there to choose 4 letters? (e) Suppose you select 4 letters. What is the probability that you have 2 “I” letters, one “M” letter, and one “S” letter? (Use your answers from parts c and d.)5. A survey of a magazine’s subscribers indicates that 50% own a home, 80% own a car, and 90% of the homeowners who subscribe also own a car. (a) What is the probability that a subscriber owns both a car and a home? (b) What is the probability that a subscriber neither owns a car nor a home? (c) What is the probability that a subscriber who is a car owner also owns home? 6. An alien rescued by the US government from a crashed UFO’s is sent to a base in either Alaska or Maine. The probability of a rescued alien being sent to Alaska is 0.8 while the probability of being sent to Maine is 0.2. It is known that a randomly selected employee of the Alaska base has a 0.7 probability of wearing a fur coat, whereas the probability is 0.5 for a Maine employee. (a) What is the probability that the first person a rescued alien sees upon arrival to a base is not wearing a fur coat? (b) If the first person an alien sees upon arrival is wearing a fur coat, what is the probability the alien is in Maine? 7. An insurance company has collected the following data on the gender and marital status of 300 customers: Marital Status Gender Single ( S )Married ( M ) Divorced ( D ) Female (F) 30 50 20 Male (F) 25 125 50 A customer is selected at random. Calculate the probability that the selected customer is: (a) a married female (b) not single (c) married given that the customer is male (d) Are marital status and gender independent? Explain using probabilities. 8. The joint probability distribution of X and Y is shown in the table: X Y 5 10 15 1 0.30 0.18 0.12 2 0.20 0.12 0.08 (a) Adding to the table above, calculate the marginal probability distributions for X and Y. (b) Find the expected value and variance for X. (c) Are X and Y independent? Explain.9. The probability distributions for X and Y are shown in the tables below: X x –4 0 4 8 p(x) 0.15 0.25 0.20 0.40 Y y 0 5 10 p(x) 0.30 0.50 0.20 (a) Calculate the expected value for X. (b) Calculate the variance for X. (c) Given E[Y] = 4.5, calculate the expected value for g(X, Y) = 2X – 3Y. (d) Given V(Y) = 12.25 and Cov(X,Y) = 14.85, calculate the variance for g(X, Y) = 2X – 3Y. (e) Find the lower bound for the probability that X is between –5.46 and 12.26. 10. The joint probability distribution of X and Y is shown in the table: X Y 1 2 3 2 0.10 0.15 0.20 4 0.30 0.15 0.10 (a) Adding to the table above, calculate the marginal probability distribution for Y. (b) Calculate P[ X = 2 | Y = 4 ]. (c) Define the new random variable W = 2X + Y. Find P[ W = 8 ]. (d) Given E[X] = 1.9 and E[Y] = 3.1, find the covariance for X and Y. 11. A random variable X has the distribution: otherwise 040for 84)(=≤≤−= xxxfX (a) Find the c.d.f. (b) Use the c.d.f. to calculate the probability that X is between 1 and 2. (c) Given E[X] = 4/3, calculate the following (Leave answers as fractions): (i) the variance of X. (ii) the expected value of g(X) = X(3X − 2). 12. Two random variables X and Y have joint p.d.f. and X has marginal p.d.f. below: fX,Y(x, y) = (3x – y)/9 for 1 < x < 3, 1 < y < 2 = 0 otherwise. otherwise 031for 613)(=<<−= x xxfX (a) Find the marginal pdf of Y. (b) Given Are X and Y