You purchase a chainsaw, and can buy one of two types of batteries to power it,

Question

You purchase a chainsaw, and can buy one of two types of batteries to power it, namely Duxcell and Infinitycell. Batteries of each type have lifetimes before recharge that can be assumed independent and Normally distributed. The mean and standard deviation of the lifetimes of the Duxcell batteries are 10 and 2 minutes respectively, the mean and standard deviation for the Infinitycell batteries are 12 and 3 minutes respectively.

Part a) What is the probability that a Duxcell battery will last longer than an Infinitycell battery? Give your answer to two decimal places.

Part b) What is the probability that an Infinitycell battery will last more than twice as long as a Duxcell battery? Give your answer to two decimal places.

Part c) You are going to cut down a large tree and do not want to break off from the job to recharge your chainsaw battery. You buy two Duxcell batteries, and plan to use one until it runs out of power, after which you immediately replace it with the second battery. How long (in minutes) can the job last so that with probability 0.75 you can complete the job using the two Duxcell batteries in sequence?

Explanation / Answer

solution=

a) Let X be the lifetime of a Duxcell battery. It is normal, X ~ N(10, 2^2) (In the standard notation that means normal with mean 10 and variance 2^2).
Let Y be the lifetime of an Infinitycell battery. Y ~ N(12, 3^2).

You'd like to know P(X > Y) which is P(X - Y > 0). You can find this from the distribution of X - Y. When you add normal random variables, their means and variances add.
So the mean of Z = X - Y is 10 - 12 and the variance is 2^2 + 3^2. That is, Z ~ N(-2, 2^2 + 3^2).

Calculate the probability that a variable with that mean and standard deviation = sqrt(2^2 + 3^2) is greater than 0.

b) Same idea. Z = Y - 2X. Since var(2X) = 2^2 * var(X) then Z ~ N(12 - 2*(10), 3^2 + 4*2^2)

c) Same idea. Z = 2X. Again by the laws of how normal random variables combine, Z ~ N(2*10, 4*2^2)

This is all about how normal random variables combine, and finding probabilities from normal distributions.

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