Suppose that the useful life of a particular car battery,...

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. Define the random variable. X = _________________________________. b. Is X continuous or discrete? c. X ~ ________ d. On average, how long would you expect one car battery to last? e. On average, how long would you expect nine car batteries to last, if they are used one after another? f. Find the probability that a car battery lasts more than 36 months. g. Seventy percent of the batteries last at least how long?

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. Define the random variable. X = _________________________________.
b. Is X continuous or discrete?
c. X ~ ________
d. On average, how long would you expect one car battery to last?
e. On average, how long would you expect nine car batteries to last, if they are used one after another?
f. Find the probability that a car battery lasts more than 36 months.
g. Seventy percent of the batteries last at least how long?

Key Concepts

Cumulative Distribution Function and Survival Probability

The cumulative distribution function (CDF) for an exponential distribution is F(x) = 1 - e^(??x), which gives the probability that the battery fails by time x. The survival probability, or the probability that a battery lasts longer than a given time, is derived as P(X > x) = e^(??x). This concept is essential for determining the likelihood that a battery exceeds a specific operational period.

Quantile Calculation

Quantile calculation involves determining the value below which a certain percentage of observations in the distribution fall. For the exponential distribution, this is typically done by setting the survival probability equal to a specified level (e.g., 0.7 for 70%) and solving for the corresponding time. This process is useful for assessing reliability and target performance metrics based on the distribution's percentiles.

Linearity of Expectation

Linearity of expectation is a principle stating that the expected value of the sum of random variables equals the sum of their expected values, regardless of whether the variables are independent. This is used to calculate the total expected lifetime when multiple batteries are used sequentially, by simply multiplying the individual expected lifetime by the number of batteries.

Random Variable Definition and Type

A random variable is a function that assigns a numerical value to each outcome of a random experiment. In the context of this problem, the random variable represents the lifetime of a car battery and is characterized as continuous, meaning it can take any non-negative real value rather than discrete, countable values.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is defined by a rate parameter (?) and has the probability density function f(x) = ?e^(??x) for x ? 0. Its memoryless property implies that the likelihood of an event occurring in the future is independent of the time already elapsed, making it ideal for modeling lifetimes such as car batteries.

Expected Value

The expected value of a random variable is a measure of its central tendency or long-run average outcome. For an exponential distribution, the expected value is given by 1/?. This concept helps in predicting the average behavior of a battery's lifetime, which is crucial in reliability and performance evaluations.

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