Question

Suppose that the useful life of a particular car battery, measured in months, decays with parameter 0.025 . We are interested in the life of the battery, (a) In words, define the Random Variable X. the number of batteries that last a lifetime the life, in months, of a car battery the number of car batteries that an indlvidual needs to replace the life, in years, of a car battery (b) Is X continuous or discrete? continuous discrete (c) On average, how long (in months) would you expect one car battery to last? months. (d) On average, how long (in months) would you expect 7 car batteries to last, if they are used one after another?

Answer 1

The random variable X represents the life of a car battery in months and is continuous. On average, one battery is expected to last 40 months, and 7 batteries used one after another are expected to last a total of 280 months.

Step-by-step explanation:

Understanding Car Battery Life as a Random Variable

In this scenario, we are analyzing the useful life of a car battery with an exponential decay parameter of 0.025. Answering the student's queries:

• (a) The random variable X is defined as the useful life of a particular car battery, measured in months.
• (b) This random variable X is continuous since it can take on a range of values in a continuum.
• (c) The expected value for the useful life of one car battery, based on the exponential distribution, is the reciprocal of the decay parameter, which is 1/0.025, resulting in 40 months.
• (d) If we use 7 car batteries one after another, and each has an average life of 40 months, then the total expected life for all 7 would be 7 × 40 = 280 months.

Note: The calculation for part (c) comes from the mean of the exponential distribution, which is 1/lambda, where lambda is the decay parameter. The answer in (d) assumes that the batteries' lives are independent of one another and the total life is simply the sum of the individual expected lives.