The life expectancy (in years) of a certain brand of plasma...

The life expectancy (in years) of a certain brand of plasma TV is a continuous random variable with probability density function
$$f(t)=9\left(9+t^{2}\right)^{-3 / 2} \quad(0 \leq t<\infty)$$
Find the probability that a randomly chosen plasma TV will last more than 4 years.

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Key Concepts

Integration Techniques

When evaluating integrals related to probability functions, various techniques such as substitution may be used to simplify the mathematical computation. These techniques help in transforming the integral into a simpler form, making it easier to evaluate the probability associated with a given condition.

Integration of Continuous Random Variables

To find probabilities associated with continuous random variables, one typically integrates the pdf over the desired interval. This process translates aspects of the distribution into probabilities, ensuring that the area under the pdf corresponds directly to the likelihood of events.

Probability Density Function

A probability density function (pdf) defines the likelihood of a continuous random variable taking on a specific value. It is a function whose integral over a region gives the probability of the variable falling within that region, and its total integral over its entire range must equal 1.

Survival Function

The survival function, also known as the complementary cumulative distribution function, gives the probability that a random variable exceeds a particular value. It is calculated by integrating the pdf from that value to infinity, representing the 'tail' probability of the distribution.

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