MATH 350 Homework 07

Throughout, please use notation like, but not necessarily equivalent to, X∼D(...)X \sim D(...) where DD is some distribution, and P[X≤x]\mathbb{P}[X \le x], as appropriate to each questions subpart. Also, please provide answers in mathematical symbols.

  1. Electrical engineers are designing a power supply unit for a data center. The time, in months, between failures of this power supply unit follows an Exponential distribution with a mean time between failures (MTBF) of 24 months.
    1. What is the probability that the power supply unit will fail within the first 6 months of operation?
    2. For what amount of time is the data center guaranteed to run with 95% probability?
  2. Mechatronic engineers are designing a robotic system, and one of the critical components is a sensor that measures the time between failures. The sensor's time between failures follows an Exponential distribution with a mean time between failures (MTBF) of 500 hours.
    1. What is the probability that the sensor will fail within the first 200 hours of operation?
    2. If the engineers want the sensor to have a 90% chance of operating without failure for an entire week (168 hours), what should the MTBF of the sensor be?
  3. Imagine you are a safety engineer at an aerospace company, and you are responsible for ensuring the reliability of a critical component in a spacecraft's propulsion system. The time, in hours, until this component fails follows an Exponential distribution with a mean time between failures (MTBF) of 1,000 hours.
    1. What is the probability that the component will fail within the first 500 hours of operation?
    2. What is the probability that the component will operate for at least 2,000 hours without failing?
    3. If the company wants to guarantee a 95% chance that the component will not fail during a mission, what should the mission duration be (in hours)?
  4. Let the random variable XX follow the continuous Uniform distribution on [a,b][a, b], which is written mathematically as X∼U(a,b)X \sim U(a, b). Calculate E[X2]\mathbb{E}[X^2].