Life Span of a Television Part The life (in months) of a...

Life Span of a Television Part The life (in months) of a certain television part has a probability density function defined by $f(t)=\frac{1}{3} e^{-t / 3}$ Find the probability that a randomly selected component will last for the following lengths of time. (a) At most 9 months (b) Between 3 and 9 months (c) Find the cumulative distribution function for this random variable. (d) Use the answer from part (c) to find the probability that a randomly selected part will last no more than 6 months.

Life Span of a Television Part The life (in months) of a
certain television part has a probability density function
defined by
$f(t)=\frac{1}{3} e^{-t / 3}$
Find the probability that a randomly selected component will last for the following lengths of time.
(a) At most 9 months
(b) Between 3 and 9 months
(c) Find the cumulative distribution function for this random variable.
(d) Use the answer from part (c) to find the probability that a randomly selected part will last no more than 6 months.

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Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time until an event occurs, such as component failure. It is characterized by its constant hazard rate and the memoryless property, meaning the probability of failure in the next period is independent of how much time has already elapsed.

Probability Density Function (PDF)

The probability density function (PDF) describes the likelihood for a continuous random variable to take on a particular value. For the exponential distribution, the PDF is typically expressed as f(t) = (1/?) e^(?t/?) for t greater than or equal to zero. It provides a basis for calculating probabilities over specific intervals by integrating the function over those intervals.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a particular value. It is obtained by integrating the PDF from the lower bound of the distribution up to the value of interest. For the exponential distribution, the CDF is commonly given by F(t) = 1 ? e^(?t/?), making it a convenient tool for computing probabilities over specified time spans.

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