MATH 314 Homework 06

Due 2026-10-14 by 11:59pm

  1. Car batteries have a limited lifespan. Let's consider a particular brand of car batteries with a mean lifespan of 5 years.

    a. What is the probability that a car battery of this brand will fail within the first 3 years of use?

    b. In how many years will 95% of the batteries have died?

    c. What is the probability that a car battery of this brand will last between 4 and 6 years?

  2. An engineering team is testing a new lithium-ion battery designed for electric vehicles. The team has determined that the distance an electric vehicle can travel on a full charge is approximately Normally distributed, with a mean of 320 miles and a standard deviation of 25 miles. Let represent the distance, in miles, that a randomly selected vehicle can travel on a full charge.

    a. What is the probability that a randomly selected vehicle can travel more than 350 miles on a full charge?

    b. What is the probability that a randomly selected vehicle can travel between 290 and 330 miles on a full charge?

    c. The engineering team wants to advertise a driving range that at least 95% of vehicles will achieve on a full charge. What driving range should they advertise?

  3. Let . A common linear transform of a Bernoulli random variable is . For instance, this transformation shows up in support vector machine models.

    a. What values are in the sample space of the distribution represnted by the random variable ?

    b. What is the expecation of , ?

    c. What is the variance of , ?

  4. Make a histogram showcasing the Central Limit Theorem for Exponential data. Let's do this in steps.

    a. Instantiate a Exponential random variable using the Scipy class expon with whatever scale parameter you want.

    b. Write a for loop of length R (> 999). In each iteration generate N random variables. Calculate a mean and store it into the rth element of a numpy array (no Python lists and especially no .append()).

    c. Outside of the loop, scale your R means using the approriate z-score calculation for your chosen Exponential distribution and according to the Central Limit Theorem.

    d. Make a histogram of your normalized means.

    e. Overlay a standard Normal distribution on this plot to highlight the Central Limit Theorem's approximation.