Applied Statistics for Business & Economics

Student ID: Name: STA 103 – Homework You will write your answers below (showing all necessary work), then upload a scan of your homework to Gradescope. 1. A firm has monitored the duration of long-distance telephone calls placed by its employees to help it decide which long-distance package to purchase. The duration of calls was found to be exponentially distribution, with a mean call length of 5 minutes. (a) What is the probability a call lasts more than 2 minutes? Final Numerical Answer Below (b) What is the probability a call lasts between 5 and 8 minutes? Final Numerical Answer Below (c) If a call has lasted 3 minutes, what is the probability it lasts an at least additional 4 minutes? Final Numerical Answer Below 1 2. In a particular state, a major flood happens once every 5 years on average. You may assume an exponential distribution. If you can assume floods are independent, and that two floods cannot happen simultaneously, find the following: (a) The probability that a flood happens within the first 10 years. Final Numerical Answer Below (b) The probability that a flood happens after the first 6 years. Final Numerical Answer Below (c) The expected time until a flood happens, and the standard deviation. Final Numerical Answer Below (d) Find the 50th percentile of the time until a flood. Final Numerical Answer Below 2 3. The wait time at a popular fast food restaurant is uniformly distributed between 2 and 10 minutes on average. (a) Find the probability that you wait less than 4 minutes. Final Numerical Answer Below (b) Find the probability that you wait between 8 and 10 minutes. Final Numerical Answer Below (c) What is the expected wait time, and the variance of wait time? Final Numerical Answer Below (d) Explain why the answers to (a) and (b) are the same. Hint: draw the probability density function for a Uniform random variable. 3 4. The total amount of tips a waitress receives is uniformly distributed between $1 and $25 in an hour on average. (a) Find the probability the waitress receives between $15 and $20. Final Numerical Answer Below (b) What is the expected tip amount the waitress will receive? Final Numerical Answer Below (c) Find the following probabilities: P r(X < µX ), P r(X > µX ) where X =the amount of tips a waitress receives. Final Numerical Answer Below (d) What is the median amount of tips a waitress will receive? Final Numerical Answer Below 4 5. A hospital receives a pharmaceutical delivery each morning at a time that varies uniformly between 7:15 and 7:30AM. (a) What is the expected time of delivery? Final Numerical Answer Below (b) Find the probability that the time of delivery is within 1 standard deviation from the mean (between (µX − σX ) and (µX + σX )). Final Numerical Answer Below (c) Find the 70th percentile for the time of delivery. Final Numerical Answer Below (d) If the delivery has not arrived by 7:20, what is the probability it arrives before 7:25? Final Numerical Answer Below 5 6. If a discrete random variable X has the following probability distribution: X P (X) -2 0.2 -1 0.3 0 0.15 1 0.2 2 0.15 Use this to find the following: (a) The mean of X and E[X 2 ]. Final Numerical Answers Below (b) The probability distribution for Y = 2X 2 + 2 (i.e, all values of Y and P (Y )). (c) Using part (b) (i.e, the probability distribution for Y ), find E[Y ]. (d) Using part (a), verify your answer in part (c) for E[Y ]. 6 7. For the following named distributions, find the value of E[X 2 ]. (a) Binomial with parameters n, p. (b) Uniform with parameters a, b. (c) Normal with parameters µ, σ 2 . 7 8. For the following discrete random variable X with probability distribution: X P (X) 0 0.2 1 0.4 2 0.15 3 0.25 (a) Find the probability distribution for Y = 3X 2 − 2X + 1 (i.e, all values of Y and P (Y )). (b) Find E[Y ] using part (a). Final Numerical Answer Below (c) Find E[X] and E[X 2 ]. Final Numericals Answer Below (d) Find P r(Y ≤ 2). Final Numerical Answer Below (e) Using part (c), verify your result from part (b) using E[X] and E[X 2 ]. 8